Everyone has their grammatical bugbears. For one of my grammar teachers, it was splitting infinitives. He could never understand why anyone, really, would be compelled to not follow this most basic grammatical rule.
But the issue that got our newspaper into hot water recently is definitely one of the classics, because it isn't just grammatical. I'm talking of the singular "they," as I used it in my first sentence. Self-proclaimed grammarians always seem to be eager to point out that using a singular "they" is not grammatically correct, but only done to avoid the unwieldy "he/she."
So let's look at the question. There are two main schools of grammar - prescriptivists and descriptivists. Prescriptive grammar tries to determine correct grammar by looking at rules, descriptive grammar does so by looking at usage.
Descriptivists have an easy go on the singular "they." Shakespeare used it many times, as did Mark Twain and George Bernard Shaw. Really, compiling a list of the authors who have used the singular "they" is all but impossible because it is so widespread. Much of this use happened centuries before "political correctness" was even a term. I'd invite anyone advocating against the singular "they" to contact the Globe Theatre in London to tell them they've misunderstood the fabric of the English language.
But what about prescriptivists? Here, the argument is more complicated. Briefly, a word isn't just "singular" or "plural." A word is morphosyntactically singular or plural, which determines how it agrees with verbs. But a word is also semantically singular or plural, which determines what the word actually refers to.
To understand why the singular "they" works grammatically, consider the sentence "The team celebrate when they win." "The team" is semantically plural (referring to a group of people, and agreeing with the plural "they"), but morphosyntactically singular (otherwise it would be "the team celebrates"). The singular "they" is semantically singular (referring to only one person), but morphosyntactically plural. The morphological/syntactical/semantic disagreement seems to be what bothers people with the singular "they" but those same people have no problem with it when this disagreement appears in words like "team," "family," or even "everyone."
In short, this isn't a real grammar issue, or a political correctness one. So why do people use a singular "they" if it gets people so riled up? Because most people, myself included, have a problem using "he" in cases when gender is uncertain - it seems to imply that women don't exist. And "he/she" or other variants are too clumsy.
But most people probably use the singular "they" in their writing because they never even considered it incorrect. Now I just have to hope none of our readers have a problem with hanging prepositions.