Learn when you should use "he," "she," "he/she," and "they." [UPDATE: The Chicago Manual of Style and the AP Stylebook changed their recommendations about the singular "they" in late March of 2017. This post will be updated soon. In the meantime, the short story is that Chicago said it's now OK to use singular "they" for transgender people, and the AP made more extensive revisions, which are detailed here: AP Style Updates.]
"He or She" Works in Formal Writing
If I'm writing a formal document, I'll use "he or she" or "him or her." For example, When you find the person responsible, tell him or her to report to my office. Admittedly, it's awkward, but if you're already using formal language, I don't think it's too distracting. (That’s also the solution recommended by The American Heritage College Dictionary.)
Can We Make Up New Pronouns?
Over the years, people have tried to introduce new pronouns such as "zie," "zir," and "thon" to fill the void, but none of these has had much success. Surprisingly, Dr. Elaine Stotko, from the School of Education at Johns Hopkins University, and her student, Margaret Troyer, reported a few years ago that school children in Baltimore were using the slang word "yo" as a gender-neutral singular pronoun--not in a way to get attention, like Yo, check this out, and not as a form of "your" as in yo momma, but like a real singular pronoun. I haven’t heard of this isolated trend spreading, but it’s an interesting development.
So, what should you do? Certainly you shouldn’t write When you find the person responsible, tell yo to report to my office, even if it’s not a formal document.
Is "They" the Future of Generic Pronouns?
I will state for the record that I am a firm believer that someday "they" will be the acceptable choice for this situation. English currently lacks a word that fits the bill, and many people are already either mistakenly or purposely using "they" as a singular generic personal pronoun; so it seems logical that rules will eventually move in that direction.