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Definitely not, using work resources for personal promotion is not a good idea.

But there's nothing wrong with including a link to your personal site on your LinkedIn profile.

Many companies have policies or guidelines regarding email signature blocks. So start there.

I would expect that even one without guidelines would not like employees to include a link to a site not related to their company or your project.

Since I use my work email to communicate with many clients and maybe
potential future employers, I would like to put a link to this website
in my email signature for work.

Don't try and talk to future employers with a email address of your current employer. They don't like that. You also don't want to lose your only way of communicating with future employers on your last day of work.

This is a decent idea if you own your own company and the email address.

If you're not a business owner and you work for someone else, this is not the best idea.

There is one exception to firm no provided by Sourav Ghosh.

If, and only if you work not as employee, but as an one-person company on B2B (business to business) agreement with your "employer" you may be allowed, or even required by law*, to disclose it. And link to your business website may be acceptable way to do it.

Always consult your contact in your "employer" company before doing it. They may have specific rules or guidelines about it.

* I am not a lawyer. If you don't know how it is for you, in your jurisdiction, you need a lawyer.

No. As an employer, the last thing I want to see is someone's personal website. This indicates 'side projects,' which we all know are important and practical, but suggests (right or wrong) that the person is working on it during office hours.

I'm not a stickler, and hope everyone has a healthy work-life balance including keeping personal tabs during office hours, but don't put it on your office signature going out to my internal and external contacts. Stop immediately is my advice to you as a member of an organization that is growing and supported by you.

I think it depends. Most of the time you shouldn't do that. Of course, if you do that, you need some kind of approval from your immediate boss (and you clearly need to separate professional matters from others). Don't do that without such an approval. Never divulge something on the Web that your employer would be unhappy of.

I am doing that all the time (with some informal approval from my boss), so this is another exception (to the accepted answer). And so are most of my colleagues, and sometimes my direct boss and my N+2 boss.

The reason is simple, but highly specific. My employer organization (the DoE equivalent in France, CEA) is somehow a research, French state owned, organization (but also the institution making nuclear warheads for the French state). For burocratic "soviet-like" reasons I won't explain in details here (but which are public and available in some Court des Comptes report written in French), we practically are not allowed to have any professional web page within CEA, that is under the .cea.fr TLD. But we all (and that includes my colleagues, my boss, my boss' boss) have some more of less "professional" web pages "somewhere" else. And we practically use them, e.g. to find H2020 research partners, to put preprint of reports, etc.

In practice, getting a web site inside CEA visible from outside is so bureaucratic (requiring dozens of internal meetings) that most of CEA web pages are obsolete, scientifically speaking. Practically speaking, there is more grey knowledge (from CEA employees, with approval from their boss) outside of .cea.fr than inside it.

But we (research engineers at CEA, LIST) nearly all have a web page somewhere, and when I did found (and that happened more than once) some H2020 research partners with my personal web page, both my boss and his own boss congratulated me.

But CEA is a special case. I hope (and I know) that most other similar government-owned research institutions in other countries are more flexible.

This is a (shameful) peculiarity of France. As a country we both love and hate bureaucracy. Why is it so it outside of my expertise. French President E.Macron nominated (with the approval of Prime minister E.Philippe) at the head of CEA François Jacq with the explicit mission to reduce that bureaucracy (inside CEA).

PS. Right now (in start of 2019), I (and my work on bismon) am funded with both the DECODER and the CHARIOT H2020 projects. And in both cases, I did found H2020 partners using a few sentences on my personal web pages, and my boss knows that, and was happy of that.