Are sign-on bonuses still a thing? If so, how do you ask for one?

Very much so. Particularly if you're switching roles a useful strategy is to mention what equity, bonuses or compensation you may be losing as a result. In my most recent swap, I indicated I was due a fairly large bonus in 60 days and would want to delay my start to make sure I earned that and quoted the approximate amount of it. The new company didn't give me the entire amount of the bonus, but definitely increased their offer to cover most of the bonus between cash and equity.

It is also useful to mention any other offers you may have and to apply for other positions, even if you're less interested in those roles. A friend of mine applied for several roles using that strategy including one he thought he'd be totally uninterested in. He wound up being most excited about that role after getting further along in the process and ultimately accepted it, even though he intended to only use it as a bargaining chip.

You also can apply for other roles and even mention "I'm early in the process, but they've stated the usual comp range is XYZ, which is higher. I'm more excited about this role, but it could be hard to turn down that kind of compensation".

As far as I'm aware - yes they are (my brother-in-law is a tech recruiter and mentions them fairly often). I just asked for one with a startup company and it was approved.

I'd just ask and see what they say?

I have a look at the contract to see what extras they want from my time not covered by normal work hours. Treat it like an options contract: if they want to option my future time, for example expecting me to do things (or not do things) after I leave their employment, they should pay for that up front.

The only thing I'd point out is that you should be sure you're willing to walk away from the opportunity they're offering you, before you push too hard. You need to be okay with them saying no to your terms.