I flew United to LA with my tenor sax and it was really easy. My first flight was from a small airport to Washington-Dulles, so I was on a pretty small plane. I gate-checked my tenor there and when I got of the plane, I made sure to stare down the airport employees to see if they were being careful. One guy recognized my stare down look and brought me my sax personally. It was super helpful and I was on my way. Then when I got on the larger planes to go to LA, I ended it gate-checking it again for ease of boarding. I had no problems with the process. I will say I had a small heart attack when I arrived at LAX because it did not come around with my luggage on the baggage claim. Instead I learned it goes to an area where it is watched by some airport employees so it isn’t stolen. The whole area is next to the baggage claim out of sight of potential thefts.

I use a protect case for reference that is both soft interior and exterior with some kind of hard frame in the middle. My horn fits very snuggly in the case and doesn’t move much. I would never use just a hard or soft case to travel I think.