Is it correct to use "their" instead of "his or her"?

Is this sentence grammatically correct?

Anyone who loves the English language should have a copy of this book in their bookcase.

or should it be:

Anyone who loves the English language should have a copy of this book in his or her bookcase.

Certainly many usage guides have advised against use of this "singular they" on various "logical" grounds. Nevertheless, singular they has long been part of the English language, and there are various posts on Language Log giving examples of it being used in the Bible, by Shakespeare, by the president, by the Canadian Department of Justice, etc.. The Cambridge Grammar of the English Language's coauthor Geoff Pullum (a frequent Language Log contributor) calls the idea that they must never occur with a singular antecedent a myth.

There is no shortage of usage "experts" who advise against it, as the other answers in the question should make clear (though these days their reasoning tends away from a simple "it's wrong" towards something more defensive–"some people will think it's wrong, so avoid it"). But despite them, use of singular they occurs at all levels of the language, both spoken and written, informal and formal.

It's not ungrammatical per se on the basis of analysis of actual usage using reasonable linguistic methods. But use it at your own risk of being criticized by the self-righteous but misinformed.

Up until very recent times the natural answer would have been "Anyone who loves the English language should have a copy of this book in his bookcase", because "his" was also a gender-neutral pronoun. It turned out, however, that "his" could only function as a gender-neutral pronoun if it were a plot by the patriarchy or something, so we're in the process of trying out alternatives.

I've heard serious proposals to substitute "their", "his or her", "her", "its", and even "hisorher". Of those alternatives, I use "their" because it sounds the most natural to me, but usually I avoid pronouns altogether, as in, "This book should be book in the bookcase of anyone who loves the English language." That style has the additional advantage of making you sound formal and pedantic.

Second one (for writing purpose), if I believe the "THEY/THEIR (SINGULAR)" article.

A good general rule is that only when the singular noun does not specify an individual can it be replaced plausibly with a plural pronoun:

“Everybody” is a good example.
We know that “everybody” is singular because we say “everybody is here,“ not “everybody are here” yet we tend to think of “everybody” as a group of individuals, so we usually say “everybody brought their own grievances to the bargaining table.”
“Anybody” is treated similarly.

However, in many written sentences the use of singular “their” and “they” creates an irritating clash even when it passes unnoticed in speech.
It is wise to shun this popular pattern in formal writing.

I have gone to using their instead of the increasingly awkward him/her in all but the most formal of my writings. His or her is sounding very contrived, and it is no better to substitute her for him than to have him as the gender neutral pronoun. The use of their is increasingly common and should simply be accepted by grammarians, though it will probably take a century or so before the high and mighty accept this increasingly common usage.