- An abstract person whose gender is unknown, or a person chosen from a set containing people of both genders. "If the reader does not like my use of singular 'they', they are invited to stick it up their jumper".
- A concrete person whose gender is unknown. "We wish to thank Reviewer 2 for their critical feedback & sincerely apologize for not having written the manuscript they would have written".
- An abstract person whose gender is known: "Just because somebody's feminine, it doesn't mean they have a vagina".
- A concrete person whose gender is known, but which you do not want to reveal. "I met someone, and they make me feel so happy... that someone is a guy".
- A concrete person whose gender is known, but does not fit within the male/female binary: "Jamie is agender; they'd prefer that you use the pronouns 'they', 'their' and 'them' when talking about them".
- A concrete person whose gender is known, binary, and obvious from context, but which you do not want to encode in the pronoun. "Tom gave me their shirt". This was the sense @cassolotl was originally asking about.
I think it's interesting that in my idiolect, singular 'they' carries an implication of abstraction or uncertainty: compare my reactions to senses 3 and 6. I'm not claiming that my personal feelings of grammatical acceptability are high-quality linguistic data, but they should probably carry more weight than a set of formal rules that were inferred originally from Latin.
A popular argument for singular 'they' is the argument from history: as one Twitter interlocutor told me, "Singular 'they' goes back to the 14th Century. Chaucer, Shakespeare, Austen and Shaw can't be wrong". But arguments from history have limited validity when applied to modern English. In the opposite direction, consider the following passage from the novel Tinker, Tailor, Soldier, Spy, published in 1974:
They had been following the coast path, each lost in his thoughts: she to Haydon, he supposed, he to Control...That's a clear usage of "he/his" as a gender-neutral pronoun in sense 1 above. Does that seem right to you? It feels weird bordering on outright wrong to me, and yet it was apparently considered correct usage only forty years ago. And if Le Carré's constructions can pass from correct to incorrect in so short a time, so can Austen et al's. Chaucer, in particular, should not be considered a reliable guide to modern English usage. But also, while I'm sure you can find examples of Shakespeare and Austen using singular 'they' in senses 1 and 2, can you find examples of them using it in senses 4-6, or even 3? I suspect not, but would welcome correction.
So, before telling me that singular 'they' is completely correct in all situations and only idiots object to it, please stop and ask yourself if you'd use it without qualm in all the senses I list above. A straw poll among a left-leaning group of friends gave me only one outright "yes", with the rest offering variations on "sense 6 is weird, but acceptable". Interestingly, there was a disagreement about whether invented gender-neutral pronouns such as "zie" were acceptable in sense 4, a situation for which I'd have thought they were custom-designed. My own view is that singular 'they', while out of fashion for a long time, is enjoying a renaissance and that all the uses I list above will be considered completely normal within a couple of decades: even though I find some of them odd, I might as well get used to them now.