Because I've just recently started shooting film again, and will leave for a week in the UK in a few days, I've been pondering that very question.
Guidance at the TSA site and at the individual airport sites I've checked is consistent: The x-rays in the machines that scan checked luggage will damage any unprocessed film. They say the carryon scanners will not. So, put film in a carryon. Then, remember that many carryon bags won't fit in the small regional aircraft, so carryons can be taken from you and stored in the plane's hold. If that happens, it's worth asking if it will be x-rayed.
The effect of x-rays is cumulative. Most sites seem to say don't worry if your film is exposed less than 5 times.
High-speed film, 800 and up, is a different matter. Even the official sites say ask for a hand chck of high-speed film.
All that said, the net is full of anecdotal reports of film getting fried in the carryon scanners, and of surly security staffers refusing hand check requests.
My own inclination, for 400 film and below, is not to trust the anecdotal reports. When someone claims an x-ray machine zapped his film, how are we to know what other hazards that film was exposed to? There are other reasons for ruined film.
Film photography is not inexpensive. If airport x-ray machines were regularly and consistently zapping film, I suspect it would be the number one topic on every photo board, as well as every travel board.
Me? Well, today, I'm inclined to take my DSLR. I plan to shoot a lot. I've got two 8-gig memory cards that will take about 1400 RAW images and it's dawned on me that i would be looking at $500-$600 in film and processing costs for an equivalent number of exposures. I'm not comfortable with that; film costs are more palatable when spread over time.