Since this is a copyediting subject, I may as well join in the fray.
I’m firmly with Matt on this one. This usage has gained popularity due to the need to eliminate the utterly silly and sexist use of “his” as a gender-neutral pronoun. Yes, you can usually rewrite sentences to get around the use of a singular “their” (the easiest way to do so is to pluralize the subject, but you can’t always do that), but the sentences often end up sounding stilted and unnatural. Using “their” to refer to a singular subject frequently results in the most natural sentence structure, which is especially important for fiction.