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

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.

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.

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.

See here or here or here.

Executive summary: His/her is to be avoided.

In a business English course at WVU in 1987, I was taught to avoid the sexist language/agreement problem by making the subject plural:

AnyoneAll the people who love the English language should have copies of this book in their bookcases.

In these days of exaggerated care for egalitarianism it is always a conundrum whether to use "his", "her" or "their". And perhaps we should be concerned about egalitarianism.

On the other hand, the traditional use of the masculine gender to stand in for both sexes still doesn't seem too outré to me. As a guy, perhaps I am missing something.

What I would do, if I were writing technical works, would be to write one with a masculine user, and the next time do it for a feminine user. Leave each work (whether an essay or a book) consistent as to gender of the subject user.

Alternatively, if you are a female writer, your user should always be female; and if a male writer, then male user.

Lot's of choices out there! Try to offend as many people as possible.

Why do we sometimes say, for example, salesperson instead of salesman/woman? That's for two good reasons:

• to avoid mentioning the gender when it's not known.
• to shorten the spoken/written thing (obviously when you use both man and woman, you are lengthening your word).

Well, the reasons for using singular-they are the same; and you can't use another pronoun better than they in this case.