Is it grammatically correct to use “they” as a singular pronoun?

There’s some critical nuance being missed in answers so far (though I strongly suspect it’s come up elsewhere here). The closest is in the sources mentioned by Mark A. Mandel, and the answer given by Matthew Carlson.

• The old use of singular they is with reference to an non-specific entity, where the use of gender would be misleading (the gender of the non-specific person is not known, and even if it is known, it is irrelevant). Hence, Shakespeare used:
  There’s not a man I meet but doth salute me
  As if I were their well-acquainted friend

  The singular they is not because the gender is not known, but because it’s an non-specific referent: not a man but = every man = any man.

• The grammaticality of singular they was disputed when the prescriptivists came to town, because “not logical”, i.e. “not how Latin does things”. It’s not how most languages do things. But it is how English does things. And it’s an asterisked logic, as language logic is.
• The new use of they as gender-neutral, and the even newer use of they as non-binary may take getting used to—although I find that hard to believe for the former (which after all, is still pretty much used in non-specific contexts). The latter is much harder, because it applies to specific referents; that’s not a “logical” constraint, but a semantic constraint. I know that I keep fucking it up in my correspondence with Sam Murray, just to namedrop. But the linguistic extension is straightforward, and it’s really a matter of familiarisation to get over the definiteness block.
• And as an English speaker, I’m profoundly grateful to those in the genderfluid community who choose to go with they. Singular specific they is nothing. Neologisms like zhe and hu: those are the real linguistic annoyance.
• And yes, if someone uses them, it is polite to respect that too. But thank you to those who use the resources already there in the language.

See Singular ‘They’ for a nice succinct summary of non-binary they.

Don’t read the comments offered via Facebook if you value human dignity. “Social engineering” my tuchus.

EDIT: Thanks to Clarissa Lohr for correcting me on specificity.