I've seen a few and that wasn't even online but in a meatspace, at a software conference. A few persons had bagdes with pronouns other than "he/she/they". Suppose that was a request of sorts. Although I haven't really talked to... err... such people ("them"? "xzem?" "xirzey'm?" what I'm supposed to use for "xe" and "ze" as a generalized group?), only saw that *ey (okay, does it work?) exist, so can't really tell more.

Once or twice. But everyone I've met who prefers Xe/Xir/Xirs will often append that with, "Or she/he/they. Fuck it, they're all wrong, use whatever makes you happy."

Yes, there are real people asking for use of novel pronouns like "xe," "sir," "hir," etc, etc, etc.

I know they exist. But how common is this? Is it actually worth spending any mental energy "worrying" about?

That's what they want you to do, because they think they're special somehow. Best is just ignore, and hopefully their house of cards will collapse at some point by their own doing.

It's uncommon. I'm not especially worried about it myself.

They is pretty novel. Yes, singular they has existed for a long time, but when was the first time anyone ever demanded that their only pronoun be "they"? I suspect only within the past 10 years or so.

That may be so, but it is a word that everyone already knows, so it seems within reason and shouldn't be hard to use in practice.

Singular they for a known person is the most confusing in practice. When I hear it I always think the speaker is referring to multiple people.

I haven't even met anyone requesting "he", "she" or "they" in real life. And if I do, I'll probably just disengage and walk away.