In light of the recent unpleasantness at LAX (never was an airport more aptly named), if you are planning to fly with a gun, I would give the matter some thought. Airports vary widely in how they treat travelers with firearms — one of the more exciting parts of the process is its complete lack of consistency — but in light of what happened, I would be hesitant about showing up with a gun case, or even a bow case.
Airport security has never been noted for restraint or common sense, and I believe that even if I showed up with St. Michael the Archangel at my right side, ready to swear that I was no threat whatsoever, I would quickly find myself either under arrest or Taken Away for Questioning, and my gun confiscated.
The answer is to ship your gun ahead via FedEx or UPS. Just about every FFL dealer has a UPS or FedEx contact and will ship it for a fee, which is no worse than the airlines charging you to do the same thing. A further advantage is that I’ve never known of a gun or gun case to be damaged by either company, but I know about plenty of damage caused by the airlines.
The best solution is not to fly at all. I know people who, unless the distance is over 1,000 miles, drive to where they’re going. It’s a grind, but since flying is now about as much fun as pulling your brains out through your nose, it may be preferable. And in any event, I have it on good authority that the major carriers will soon be charging for the air they pump into the cabin. If you pay a fee, the mask drops and you get to breathe. If not, you’re on your own, and they hope that you enjoy your flight and look forward to seeing you again.