The other 'not-so-obvious-place-that-requires-a-visa' is Australia. AA has always checked my visa when leaving the U.S. Turn up in Oz without one and it's probably the next plane back for you and a $5,000 fine for the airline.
ETA (Electronic Travel authority ) used by Australia is the best system I have seen. Just make sure you apply for one before you arrive at the ticket counter , as only a few agents have figured out how to use the ETA system to get one. The instructions are a real nightmare.
At the airport the agent electronically verifies the validy of the visa and the ETA system responds with an "ok to board " message, or on bad days "no eta found do not board" .
The problem I have found with other visas is that sometimes you cant read the handwriting on some of them, and some of them can be changed by hand.
Just to add, most airlines use a system called TIMATIC. Some airlines have it online . Verify your information , print it off and take the information with when you check in. Not everyone always reads the rules and requirements. Saves any headaches for you.