The issue: What pronoun do you use in a situation where the gender of the person acting either isn't known or isn't relevant? For example:
When an employee arrives . . .
The rest of the sentence is about the employee, who must sign in. What pronoun do you use?
When an employee arrives, ____ must sign in.
At one time, writers simply used he as the generic pronoun:
When an employee arrives, he must sign in.
But eventually came the complaints of sexism. (What if the employee is female?) That's when we started seeing a lot of he or she, just to be sure we covered our bases:
When an employee arrives, he or she must sign in.
That's seriously clunky and awkward, but it's better than the other weird compromise, s/he.
Others have opted to use she instead of he. That's annoying to me as a reader, because a) it's reverse sexism and b) historically he has a far more neutral feel than she, which jumps out like a flashing red light.
(Good writing should move smoothly, without jolts or flashing red lights.)
To keep the gender thing fair, some writers alternate between he and she throughout a piece. Personally, I think that goes beyond annoying and enters the range of shoot me now.
I've seen magazines that alternate on an article level: this article uses he, and the next one uses she. Not a particularly elegant solution, but at least it doesn't have me wanting to hit something.
So the gender-neutral problem persists: English simply doesn't have a singular, gender-neutral pronoun.
Finnish does have a gender-neutral pronoun, and I have to say, it's really convenient when you see a baby but can't figure out the gender. You can totally compliment the kid without offending the parents. Too bad English doesn't have an equivalent of hän.
(Another side note: Finns often use se instead of hän . . . which means it, even when referring to people. Totally works in Finnish. Not so much in English. Can you imagine referring to your friend and saying you're going to lunch with it?)
Chicago and a lot of other style guides suggest avoiding the problem altogether. Either 1) reword the sentence so you don't need the pronoun, or 2) change the sentence so you can grammatically use the plural:
When employees arrive, they must sign in.
That works fine at times, but it's still not a solution. Sometimes a piece needs the singular, and making it plural or otherwise doing acrobatics to avoid their as singular sounds odd.
This is precisely why their is becoming increasingly accepted as the singular pronoun, at least in conversation and informal writing. I'm in the camp that accepts this usage already (obviously), although some people still foam at the mouth when they see their used this way. (Just as I foam at the mouth at infer used for imply and other losing usage battles.)
That said, if I'm writing for a professional journal or something similar, I avoid using their as a singular. You write to fit the register you want the piece to fit in. If something isn't accepted in that arena, don't use it, and no, their is not accepted as Standard English.
I believe it's just a matter of time before their is considered correct and perfectly fine to use this way. People already do, often, sometimes by accident and other times absolutely on purpose (raising my hand here).
The new rule actually reaching style guides? That may take some time, but it'll happen.
Grammar Girl agrees with me and adds that "it takes a bold, confident, and possibly reckless person to use they with a singular antecedent today."
What can I say? I live on the edge.