I've flown back and forth across the U.S. several times this year carrying a bass. I've flown United, American, and US Air (that I can think of). Each time, I have just carried the bass on - usually in a hardshell case. Most airlines, including all the ones I've flown, have an exception to the size limit in their carry-on luggage policy that allows musical instruments.
I have successfully put the bass in its hardshell case in an overhead bin once, and in the plane's coat closet several times.
On one set of flights, the planes did not have a coat closet or overheads big enough to put the bass in. On those, I gate-checked it. That means I got to the doors of the plane and they put a tag on my bass case and put it in the cargo hold. Then, as soon as I got off the plane at the end of that flight, I got my bass back, right outside the doors of the plane, before proceeding into the terminal. Even in a case (no pun intended) like that, I don't really have any concerns about the bass getting damaged or lost.
It definitely helps if you are in the priority boarding group(s). Being one of the last people on the plane is a good way to ensure you have to gate check it because the overheads and closet get full quickly. I haven't tried it myself, but one thing I would consider trying if I needed to in the future would be to ask the gate agent if I could be moved up to a priority boarding group - like they do with people traveling with children or people in wheelchairs - with the explanation that it would really help me to be able to find a place to stow my guitar. Sorta like traveling with a child...