Traditional carry-on X-ray machines: film ISO 800 and slower rarely shows visible damage after a few passes - but damage accumulates, so be careful on multi-leg trips.
CT scanners (the newer machines rolling out worldwide) use far stronger doses and can fog film with stripe patterns in a single pass.
Checked-baggage X-rays are the strongest of all - never put film in checked luggage.
Best practice: carry all film in a clear plastic bag and ask for a hand inspection at security. Most airports will accommodate: "Film, hand check please."