I agree with beefheart.  I think the best practice for an essay or article is to use "he" or "she" in a random order in different segments of the essay, thereby avoiding the use of "their", and, at the same time, negating sexism.

Example:  "When the surgeon encounters unexpected cancer she should call for the pathologist.  Within 15 or 20 minutes of the pathologist's arrival, he can provide a frozen section diagnosis of the cancer type."

It's easy to fall into the trap of "their" when "her" or "his" is appropriate. I generally will write his/her the first time and then just settle in with "his." This is also what I do for "they"—when the singular is called for. So I use he/she the first time and then adopt "he" for the remainder of the paragraph. Being "politically" correct is all well and good, but sometimes I feel that we spend too much time in observing this "rule," and the content of our writing begins to suffer. I don't wish to offend "he" or "she," but I can't "he/she" throughout my writing with each reference without feeling like the content suffers. I hope the "pronoun gods" will forgive.

When I see it hasn't been fixed in someone else's writing, it makes me a little crazy, but I can easily understand how it can happen. We just need to proofread more carefully and maybe add this element to the things we check all the time, like sticking to the same tense throughout a writing.

Well the interesting thing is that the use of plural their in place of singular s/he or his/ers etc is not something to be "fixed." It is now something that may be cultivated (by choice, not mandate of rule) as a viable and workable option to the new non-gender requirements of writing. So when the University of Manchester, UK, wrote their, they meant their.

I'm still quite young to this profession but I consider myself old fashioned when it comes to improper pronoun usage.  The only time I accept the use of "their" over he or she is verbally, when mentioning the gender in question could compromise an identity that is supposed to remain anonymous.

To me, it is never acceptable in writing, and only minimally acceptable in speech.

Along the same lines, my blood boils when (especially common in interviews with professional athletes and entertainers) a person talks about his or her experience and continually speaks using the 2nd person "you" instead of simply saying "me," or "I."  Example:

"How does it feel to be the most well loved female in pop culture?"

"Well, at first you feel a little overwhelmed.  You're thinking, wow, this is a big responsibility.  But then, you just sort of get used to it, and you start to understand that your talents and your fame can be used to change the world, you know?"

(Meanwhile, I'm screaming at my TV, "Me?! My talents or yours, lady?!")

It's easy to fall into the trap of "their" when "her" or "his" is appropriate. I generally will write his/her the first time and then just settle in with "his." This is also what I do for "they"—when the singular is called for. So I use he/she the first time and then adopt "he" for the remainder of the paragraph. Being "politically" correct is all well and good, but sometimes I feel that we spend too much time in observing this "rule," and the content of our writing begins to suffer. I don't wish to offend "he" or "she," but I can't "he/she" throughout my writing with each reference without feeling like the content suffers. I hope the "pronoun gods" will forgive.

When I see it hasn't been fixed in someone else's writing, it makes me a little crazy, but I can easily understand how it can happen. We just need to proofread more carefully and maybe add this element to the things we check all the time, like sticking to the same tense throughout a writing.

When I first encountered "their" in singular usage, it was some years ago on my daughter's university information. She was preparing to study for her master degree at The University of Manchester in the U.K. In the housing etc information was if you are the student, the student this the student that and have their .... I couldn't believe it. It is imprinted in my psyche and physiology as an incredibly bizarre moment! Lo these many years later, I begin to see the value of it though ... begin to see.

Call me old fashioned, but I cringe when I hear anyone mix singular and plural pronouns, including the use of "their" for a single person.  Antiquated and cumbersome as it may be, I insist that my students use the generic "one" if they are unsure of gender. Thus, if one turns in one's homework, one will not be disappointed in one's grade. It works for me!! 

i have noticed a recent trend of alternating in academic writing and in some kinds of non-fiction books for general audiences.  For example, the author will write a paragraph using a female to make a point, using only the female pronoun, and then another paragraph using a male to make a point, using only the male pronoun.  I rather like this, but it doesn't always suit the content.

I agree that it is a function of one's audience, and I am a purist when it comes to pronouns and their antecedent nouns for academic or other kinds of formal writing.  However, I sometimes will use "their" when I am speaking, although I usually don't in front of my students.

I do think that eventually, "their" will come to be acceptable in English or someone will invent a gender-neutral singular noun that will catch on.  It is really a lack in our language, much as "you" being singular and plural results in our trying to create plural forms like "you all."

It all depends on my audience. In formal, scholarly writing, I generally use "he" rather than the unwieldy, cumbersome "s/he." For informal writing, I use "their" because it's common and easily understood. In conversation, I rarely use anything but "their." I feel no sense of outrage over gender inequity in this regard.

I use the pronoun consistently when referring to an unbiased gender in writing. Historically, the pronoun is meant to be used to represent the plural, but given that gender equality in writing has become such a predominant criticism, the use of their simply seems to be the only way to solve the problem.

Their in the singular goes against many years of conditioning and training in grammar--it is anathema to any traditionalist!

The way to avoid the he/she issue is to make general statements plural.  For instance, rather than saying or writing,

If a person is to possess integrity, he/she must have the attributes of honesty and fortitude.

one can say or write,

If people are to possess integrity, they must have the attributes of honesty and fortitude.

These politically correct concessions of avoiding he or she are part of the weakness of American culture and the corruption of the English language.  As Scrooge would say, "Bah! Humbug!"

Sined,

A traditionalist

I just find "their" highly overused.  It leads to sentences like: "They didn't have their property rights so they sued them for their money" and lets students off the hook for being specific.  I do prefer he/she following proper nouns in a sentence.  There's an equal epidemic of saying "there" and they're" as interchangeable with "their".  Just a pet peeve of mine.

I have no problem using "s/he" - and do so when needed!  "His/her" is harder to shorten, unfortunately.

I'm a bit of a fanatic about trying to make my grammar both correct and inclusive.  I appreciate the intent behind using "their" in situations as described above and understand why folks are searching for an easy way to communicate without upsetting readers of unknown sex, but it doesn't work for me.

I don't consider it specifically correct grammatically, but I agree with pohnpei in that it solves the gender equity problem. It sure solves the problem of having to use the slash: "his/her," "he/she."

It still does seem a little strange to me, but it has come to seem less strange than simply seeing people use "he" the way things were in the past.  Sure, it's grammatically incorrect and that's annoying.  But I think that grammar evolves and this is an example of that process in action.   I'd say that it's better to be technically incorrect grammatically than to use words that imply that being male is the norm.

As far as other ways to accomplish the gender-neutral thing, "their" is so much less cumbersome than saying "he or she" and then having to follow up with "his or her" and so on.  So I see it as an acceptable shortcut that gets the point across.