Would it be illegal for the UK to negotiate a trade deal with the EU while still a member?

Or is it the signing of trade deals that is prohibited, and negotiation is permitted?

Illegal? Probably not. Remember, the initial plan was to invoke article 50, and spend two years on hammering out the withdrawal agreement and the "future relations". A trade deal between the UK and the EU falls under the future relations.

But the EU has made it clear it won't start negotiations about future relations before there is a withdrawal agreement. As of this writing, it's 70 days before the 2 years are up. There is no withdrawal agreement yet. Hence, at this moment, the EU will not negotiate a trade deal.

The UK could negotiate a trade deal, but other countries aren't really interested in putting resources into a negotiation that will probably take several years anyway at this stage. Until the UK has decided what its future relationship with the EU will be the parameters of any other trade deals are impossible to define, so any talks would be extremely preliminary at best.

It wouldn't be illegal for the UK to do, but it wouldn't be of much use either. The signing of such deals would be problematic at the moment because they would conflict with the treaties that established EU membership.

Together, the UK and the EU27 could change any EU rule.

The EU decided that EU members cannot negotiate trade deals with third parties while they are EU members. Those rules did not go to much length when it comes to EU members who have negotiated and ratified a withdrawal agreement under Article 50, and either way the EU would not be a third party. EU members can and do negotiate matters affecting trade with their fellow members.

They cannot change logic:

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  The UK cannot sign trade deals with a third party while it is still an EU member. Negotiating them may be a gray area, but see the next point.
  
  That's part of the EU rules, applicable as long as the UK is in the EU. The EU could easily change the rules to permit any member to negotiate trade deals as long as those deals only come into effect if and when the member leaves, but which third party would bother to make such deals?


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  The UK cannot have meaningful negotiations with a third party before the Brexit conditions are clear and at least an outline of the post-Brexit relationship with the EU becomes apparent.
  
  Of course they could start negotiating sooner, but they would have to stay terribly vague. Say they negotiate duty-free trade in lawnmower parts with China. The Chinese would naturally want to know if they can assemble the duty-free lawnmower parts in an UK factory and sell them duty-free to the EU27, or what the duties would be in that case. And the EU27 would not agree on duty-free trade in lawnmowers unless they know what treaties with third parties the UK got in this regard.
  
  That's not in any rulebook, just common sense. They could decide to negotiate a comprehensive trade treaty with, say, the US before they negotiate one with the EU, but only if they crash out of the EU on a no-deal basis first. Then the deal with the EU would be constrained by the terms of the US deal, and not the other way around.


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  The EU27 want to negotiate Brexit first, then the Post-Brexit relationship. In theory, they could negotiate both simultaneously. That would greatly complicate the issues, but it might also open new chances for compromise ("let's make a deal, I get full market access in this area if I pay that percentage of MEP pensions for the next 30 years").