Remember the story of Rudolph the Reindeer

That's not the name he's usually known by though is it? He has an extra descriptor, that being the nature of his being different from the others. The moral of the story of Rudolph is quite simple.

Deviation from the norm will be punished unless it is exploitable.

You've said it yourself, any minority group singled out for special treatment will come to a bad end. Why are you trying to register them? Because little Johnny whose nails grow slightly faster and stronger than normal might be a danger in the future. How is sticking a label on Johnny now going to do anything other than mark him as "other", ruin his employment chances, and drive him towards the end you're trying to avoid?

There is absolutely no way anyone with a basic knowledge of history is going to cooperate with your registration program.

Your solution is to keep everyone ignorant of the outcome every previous such program has had. Keep the population blissfully ignorant and they might just toe the line.

Benefits

In today's economy, all you need to do is providing free health care and free tuition for college/university. Some tax exemptions will go a long way towards that too. Also provide tax exemptions to companies who hire them.

Instead of mutants trying to hide from society, your problem will be that you will have millions of people pretending to be mutants.

"So what is your super power?"

"I can bend my thumb backwards, look!"

"We've been through this before John, impersonating a mutant is federal fraud."

"Come on sarge, it's the only way I'll be able to treat my diabetes and/or get a diploma!"

Just register everyone.

Make being registered the prerequisite for participation in parts of society.

Escalate the amount of data collected and the parts of society for participation in which registration is required. A possible order could be:

1. Start with name, place and date of birth, names of parents, place of residence. Make it mandatory for things like voting, running in elections, serving in the military, social security...




2. Add more data with specific purposes. Blood type, specific DNA sequences, etc. for getting health insurance.



3. 
   

   Add more purposes, without adding data. Now you'll need to be registered to get a bank account. Or to get a scholarship. Or to get into university at all.

   
   
   

   a. Add even more purposes. Employment, taxes, etc.

   
   



4. Include full DNA sequence into the dataset. Find an excuse, like curing cancer.




5. Make registration mandatory.




6. Optional: Exclude unregistered individuals from civil rights. Later: human rights.

Don't make it mandatory. Do incentivise it. And heavily punish use of powers for crime.

Things like a government-sponsored training and health programme, to help people adapt to their abilities, and manage any resulting health issues (specifically ones relating to their powers, not general healthcare) which is available to registrants.

You can also set up special "fast track" schemes for certain classes of powers:

• You can control and extinguish flames? The Fire Service would love to have you


• You can teleport yourself and up to 5 other people? The President's security detail has an opening


• You can make people buy just about anything you touch, regardless of how bad it actually is? That's not a real power Mr [Gates/Jobs] (delete as preferred)

Makes sure that anyone who uses their power to commit a crime is punished severely (since it can't be confiscated like a weapon), but especially so if they were not registered. (N.B. Do not punish them more merely for having powers and committing a crime, only if they use them to perform the crime. The power to breathe underwater has no bearing on an armed bank robbery in the middle of a desert, for example)

But, really: Your government does not need a registration programme to be able to protect its people. In fact, speaking as someone who used to work in a governmental Emergency Planning department, over-reliance on such a programme would actually reduce your ability to do so.

An overview of the sorts of powers that might be out there (and any powers that the Emergency Services have on record from their staff) will allow for the production of generic Emergency plans for a wide range of situations, instead of over-specific ones that are too limited in scope to be of any use.

Register each other.

“Under the spreading chestnut tree I sold you and you sold me:

There lie they, and here lie we

Under the spreading chestnut tree.”

George Orwell, 1984.

People who know the mutants know what they are. A government which rewards cooperation and punishes recalcitrance will have some of the people all of the time and all of the people some of the time, but you really only need all of the people once to accomplish this end. This is the tack taken by the government in 1984: persons were asked and sometimes compelled to inform on each other.

The Carrot...

Offer your registered mutants things they want or need.

• Shelter


• Health care


• Education


• Powers Training (so their out-of-control powers don't kill all the people)


• A sense of community / belonging


• A guaranteed mutant income (provided they don't violate the rules...)


• Get celebrity mutants on-board as sponsors / Public Service Announcement faces.

...and the Stick

Threaten harm to those who refuse to register

• Loss of job


• Loss of freedom / mandatory jail time


• Military conscription if their power is strategically or tactically useful


• Make registration a requirement for employment for mutants, like a work visa for immigrants


• Make registration a requirement for university enrollment


• Require registration for any kind of required certification or license (doctor's license, bar exam for lawyers, etc.)


• Require registration for vehicle ownership / insurance


• Require registration for any kind of government ID


• Show stories of non-registered mutants committing crimes to spin up the negative public relations, convincing mutants that only the bad ones don't register

But be careful

You will need to have a strong public relations / marketing group work alongside the pro-registration lobbyists. There will be extreme push-back from civil liberties groups like ACLU in the USA. There will be constant comparison to the Jewish registrations of Nazi Germany, and for good reason. Whether your end-goal is the same as Nazi Germany's or completely different, you're going to have to convince the public -- and the mutants -- that registration isn't a step towards genocide.

And if your end-goal is, in fact, mutant genocide, well... that's bad. And your mutants -- registered or not -- will fight back.

You don't make it official

Simply put, you don't register them officially, there will be no government mandated registration or anything of the kind, instead you use data sharing to single out potential mutants.

Say, for example, if the mutant's father took one of those DNA tests to discover his ancestry and the latent mutant gene was flagged on the test, the government would be immediately allerted and this person would be put in a secret watching list.

Someone put photos on their Facebook that has gone trough some AI analysis thing-magig and flagged as a potential mutant? Go to the list.

Someone has recently bought a spandex suit and custom ruby-glass glasses on Amazon? Go to the list.

Maybe a mother has googled "I think my daughter is a mutant", now her daughter is going to the list.

Data is being collected all the time from all kinds of sources and this will only increase on the future, the government won't have problem accessing it and using it.

From them on the government can use the excuse of national security to scrutinize these people's entire life just to make sure they are not one of the "bad ones".

If you are on social media you will see that many societies interested in accessing the users' personal data and information do not ask directly for "give us your details and your connections", but rather develop a rather dumb game (i.e. what would you have looked like in the Aztec Empire?) and, with the excuse of sharing the results with the contacts on the social network, ask the user access to those data.

Do the same: create a fictitious game which easily give a rewards, ask the participants to provide some personal data for enrolling, one of them incidentally being "which kind of mutant power do you have?", and let the masses play.

Require that they allow their powers to be studied, but let them do so wearing a mask

Whatever plan you make to respond to a known mutant becoming hostile will also work against an unknown mutant becoming hostile, so you don't need to track individuals. What you need is to be able to respond to a hostile mutant. After all, a hostile mutant could just spontaneously show up since as you said:

These abilities manifest at different points in life, some at puberty, others well into middle age.

So new mutants can appear that you don't know about at any time. You'll always have some unregistered mutants.

To make response plans to contain or apprehend hostile mutants, you need as much information as possible about their powers.

So here's your pitch to the mutants: They have to anonymously call the FBI and say they are a mutant, and set up a meeting. They have to meet with government scientists and demonstrate their powers. They can do this part wearing a disguise if they want. They have to let the scientists take video footage and collect sensor data and other tests that don't violate the person's body autonomy. The mutant person can then leave and remain anonymous. Law enforcement will review the tapes and data they got and use it to prepare for the possibility that such powers are used by a hostile person.

This is easier for the mutants to accept because they get to remain anonymous. They aren't on a list that might maybe later be used to round them all up and put them in camps. They don't have to disclose who they are, just what they are capable of.

So mutants get to remain anonymous, but law enforcement has as much information as possible to help them prepare for hostile mutants.

How is this enforced? Simple. When a mutant comes in, their powers are added to a database. It's a crime to have powers that aren't in the database without reporting them. So if investigators find that a person gained new powers that aren't in the database, but didn't immediately report it, then that person is guilty of a crime, something like 'Failure to Report Novel Mutant Abilities". Mutants will anonymously report their abilities to avoid getting in trouble for that.

Transform them in Heroes

Heroes, Idols, Actors, Politicians, etc...
Lets take Captain America for example: in the Marvel universe there is a lot of prejudice against mutants. Captain America being a "super human" could be labeled as one for a ignorant person. Yet the majority of population do not, because they treat him as a symbol of their country. Government made him famous through media propaganda even before he went to war and actually fought.

You just have to make a propaganda that mutants are cool. But it need to be fast before the population starts prejudice, because racism is really hard to get rid of.

Create a private company, Supers, that's like Facebook but only for mutants. Advertise well. Make it seem really fun, and exclusive; in order to join, people have to demonstrate (via video, DNA test, or some other method) that they've got superpowers, and make sure only other members (and you, of course) can see all of the activity on the site. Encourage people to invite their super-powered friends and colleagues. Leak the occasional snapshot of a particularly cool post so that people get reminded of how awesome your site is.
Congratulations! You have a mutant registry.

Require a license for extraordinary-skill employment...

Simply, mutants will register because they want to use their mutant powers to get stupid rich.

...and then, create extraordinary-skill jobs.

That is to say, do the exact opposite of a school-prison pipeline where at a whole-society level you deny them any possibility for success, expect and watch them to fail, and then take their freedom or their life when they do.

Find work and meaning for them.

Take Quake, on Agents of Shield. In my world, they send her to school as a geologist. She's seem walking all over earthen dams with a can of orange spray paint and an "experimental ground-penetrating sonar that only she knows how to use" as theater to hide her abilities. She makes a fortune doing this, and even more as her company finishes a canal to link the Caspian sea.

Of course, some will go in other directions, the way Arthur Petrelli didn't really use his flying skill to run for President.

Quote from asker's question: "Historically, singling groups out to be labeled for these kinds of reasons has ended badly for those people."

If we assume that it was just singling out those people that we assume caused them grief and completely ignore the fact that an extreme amount of racism was the reason for singling them out, then you are missing a major flaw in that reasoning. If you single out group X, then you also single out group Y which is everyone not in X. Now if singling out a group causes one of the two groups to be pushed down and oppressed in some capacity or discriminated towards, well then you're betting on the wrong group being the one to be pushed down in this situation. See, with historical examples some of them had people in equal positions of power. For instance, in the rise of Nazi Germany there was political power but people were still all of equal ability. We know for a fact that Hitler had not somehow perfected some advanced race that was actually superior to everyone else. That was either a delusion or a convenient excuse to gain support for his rise to power. Now on the other hand if one considers the African slave trade that led to most if not all of the slave population in the early US, then in that situation it wasn't just political power. The Europeans had guns. The Africans didn't. Guess who won out?

If you see where I'm going with this, the registry won't result in discrimination. If anything it will just serve in lighting a powder keg. You have people that can with little to no effort demolish a city block. Do you honestly think that if discrimination occurs that eventually turns slowly over the course of decades into economic oppression and then eventually to slavery that the people with actual physical and mental superiority with potentially deity-like abilities will be the ones to be oppressed? If so, then the government would need a plan in place to capture these people immediately. No the truth is that if anything a growing population of such people will become in charge. After all, imagine a politician that can read the minds of anyone in the room or subconsciously influence their opinions? They will automatically be able to win over any crowd. Imagine someone with the strength of 20 men working in construction. They no longer have to worry about heavy machinery and the extra overhead of maintenance. All you need for that person is some metal protective gear to make sure they don't injure themselves. The list goes on. If anything, those people are likely to put everyone else out of business. And sure there will be criminals, but when there have been disproportionate numbers of a certain group being criminals that was due to them already being in certain economic situations due to past discrimination. There is no conclusive evidence that any particular race is more likely to be a criminal. The same applies here. I'd expect the same distribution of mutants to be law abiding as everyone else.

So how do you convince everyone to do a registry?

Don't call it a registry. Call it an upper society pass, because eventually that's what it will become. You'll need one to get into restaurants, get certain jobs, maybe even vote, etc. Those without powers will eventually become the minority and unfortunately will eventually naturally die out as the population mixes and those with powers and non powers inter-marry. At that point a registry has no more danger than putting your race on a census, because literally everyone will have powers and not having powers will be unheard of. Unless you are a walking nuclear bomb, nobody is going to think anything of you being super strong like that other guy who lives 3 blocks down. I mean sure some may take offense to listing powers on the census, but it's not likely going to lead to anything more than remarks of "this seems like a stupid question to put on a census, but oh wait it's a census and they want demographics".

The way I see it, there is no one thing that will solve this issue. It would require great effort and investment from a government to achieve this. It's kind of an utopia.

1 - Education

First of all, people should know what mutants are. After they've singled out what is it that makes someone a mutant (a term that already suggests that the difference is probably genetic), this should be public knowledge.

Kids should learn in school that there's nothing wrong with being a mutant, that they're part of society. If there are bad mutants, they're not bad because they're mutants, they're bad because they're people who made the wrong decisions.

This comes as point 1 because it's the most important point. Only if everyone understands collectively that mutants are not a reason to worry can this society stand.

2 - Laws against discrimination

There should be rough legislation to prevent people from discriminating anyone for being either a mutant or a non-mutant - it should be equal. A cirminal is a criminal, regardless of his genetic heritage.

Companies should be obligated to hire proportional amounts of mutants and non-mutants to companies, and there should be no difference in benefits, salaries and anything else that might be used as an argument for discrimination.

Media should also be regulated in this aspect, in the sense that the very channels of entertainment (TV, movies, etc) should depict mutants and non-mutants equally.

It's also worth mentioning that non-mutants are already regulated. Everyone is obliged to always keep their ID's with them at all times. The only difference would be that the mutant's ID would have a few additional information, like their super power. If a non-mutant is caught without an ID, he should be liable to the same consequences as a mutant.

These laws should be harsh in the sense that whomever discriminates anyone, be him/her a mutant or non-mutant, should suffer very serious consequences (fine, jail time, etc).

3 - Capable Law Enforcement

Since there's no way to predict what a mutant's power might be, the whole law enforcement system would have to be reworked to be able to deal with super powered humans.

Police stations should have the necessary gear to deal with emergencies and combat-skilled mutants should also be an expressive share of the police force, to deal with a mad dude who can blow shit up with his eyes.

It's worth mentioning that the very law enforcement should also be under the scrutiny of point 2, in order to prevent the abuse of authority.

4 - Super Jails

Having a structured law enforcement is useless if you can't keep those guys out of the streets.

The feeling of impunity is a spark to discrimination. If a government continously fails to give a sense of justice and security to society, it will eventually be the spark to the creation of hate groups and such - which we don't want.

The government should have R&D centers dedicated exclusively to the development of new technologies to deal with mutants, both for the front (law enforcement) and the prison system, which would basically have to be custom made to the kinds of offenders that eventually show up.

You can't standardize jails. Or do you think that the same structure would be capable of locking up Superman and the Martian Manhunter?

I had a lot more ideas for this but I got carried away writing and forgot the rest. If anything else comes to mind I'll add it here.

Nevertheless, I think this post was enough to give you a glimpse of what this society should be: informed and amicable towards the differences - but harsh and equipped enough to punish whomever tries to crack its foundations.

Don't register all mutants, register just the bad ones. Here you're doing something akin to the Sex offender list. Everyone on that list is a sexual deviant, but not every sexual deviant is on the list. The ones that do not break the law are not required to register who they are and what they do. Only those found to be bad by due process are registered.

Or the no fly list. You're again not registering all people, just the ones who have shown evidence that they are likely going to commit terrorist actions. This one is a little odd because this is more a "watch out for this guy, we think he might do bad things" rather than "watch out for this guy, he's done bad things". Thus, you cannot place legal restrictions on a person on the No Fly list because the accused has a right to defend himself. However, private companies are allowed look at the list and will not do buisness with them as there is to great a risk to them. This allows for bizzare situations where you may not be allowed to fly, but you may buy firearms. Again, the later is considered a right of the people, while the former is not infringing on the listee's rights because it is a specific ban on transit options, but he still may travel across state lines.

Basically, focusing on all mutants will only make otherwise good people fight for their rights (which turns them into bad actors in greater numbers). A good person does the great-power great-responsibility thing is going to get a little angry that he's still considered a monster by the law. It's one thing when its a lone journalist who was clearly writing "Fake News" before it was cool (especially if he doesn't know that the guy who gets photos of mutants is himself one and thus, making bank off of you) but it's something else when the government, which can take your rights away, does it.

So the solution is only register the obvious bad actors. If you're a good mutant, you have nothing to fear from a list of bad people, right? Right? (Hint: It's probably fair bet that anyone who justifies a policy like that will eventually expand the scope to include the people who shouldn't have feared it).

You want to keep bad hackers mutants out of your way? Hire good hackers mutants.

No real hacker would sign to any "hacker's list". So what you do is actually hiring as many as good as possible hackers to defend from bad hackers. This is the reality. Don't expect mutants to be any different. You may capture some, but you'll never really all of them registered. So just be prepared for the potential mutant's blow.

You may even use similar approach to discover those mutants - create some cool, supper difficult (in fact impossible for non-mutants) tasks that are rewarded well. Break into underwater vault, return a flag from a top of a mountain, Get to a liquid lava surrounded terrain...

Even if some non-mutant manages the task, hey - he did an excellent job and can be useful too!