Consider this from Shakespeare’s A Comedy of Errors, Act IV, Scene 3:

There’s not a man I meet but doth salute me
As if I were their well-acquainted friend

A lot of people would declare that his should be used instead of their, including some self-proclaimed grammar experts (I call them grammar terrorists).

I have been using the singular they occasionally, especially when I don’t want to specify the gender. For example, I might write:

Someone came up to me. I didn’t like the look on their face.

instead of:

Someone came up to me. I didn’t like the look on that person’s face.

In this case, there isn’t much difference. But if I continue on the narrative, the use of that person can become more cumbersome. Consider:

Someone came up to me. I didn’t like the look on that person’s face. Nor did I like the person’s smell, as I soon discovered.

which doesn’t read as smooth as:

Someone came up to me. I didn’t like the look on their face. Nor did I like their smell, as I soon discovered.

Don’t like it? Let me quote from Language Log (Shakespeare used they with singular antecedents so there):

By all means, avoid using they with singular antecedents in your own writing and speaking if you feel you cannot bear it. Language Log is not here to tell you how to write or speak. But don’t try to tell us that it’s grammatically incorrect. Because when a construction is clearly present several times in Shakespeare’s rightly admired plays and poems, and occurs in the carefully prepared published work of just about all major writers down the centuries, and is systematically present in the unreflecting conversational usage of just about everyone including Sean Lennon, then the claim that it is ungrammatical begins to look utterly unsustainable to us here at Language Log Plaza. This use of they isn’t ungrammatical, it isn’t a mistake, it’s a feature of ordinary English syntax that for some reason attracts the ire of particularly puristic pusillanimous pontificators, and we don’t buy what they’re selling.

I wish I knew this stuff when I was in secondary school - I definitely would have used the singular they in my next English composition, and hope that my English teacher would fall for my trap and put a bright red circle around my carefully-placed they, then we could have a heated argument where I would quote Shakespeare and the rest, and finally prove to them that I was indeed correct (as usual), and thus obnoxiously show my superiority over them.

Sometimes I wonder why so many of my teachers couldn’t stand me.