I think the TSA is at least the biggest place to check. Back in the early 2000's (but definitely at least a few years after 9/11 so it was in the newer rules) I remember reading good advice in "Acoustic Guitar" magazine. They had a letter from the then head of the TSA, where it stated that in all cases musical instruments such as guitars would be accommodated, a directive to in most cases allow for carry-on of guitars, violins, etc.
They recommended keeping a copy with you (in the guitar case maybe) and presenting it if you meet resistance on their part to allow. They did point out that at least at the time, employees of TSA were not consistent, and in many cases ignored the letter but it was one thing that might help. You do want to be polite.
But the best advice was to use the letter if at check in they tried demanding (TSA or the airline personnel) you check it in, ask to please be allowed to try to carry on, and mention that you would understand if cabin crew could not accommodate you and you won't make a fuss.
The idea is, let the cabin crew decide. If you got that far, they might be (or not) more inclined to try and help, but even if they tell you it has to be checked, the good thing is, it is taken from you there at the ramp by a baggage handler, that will put it on top of the rest of the baggage (and often they say in a "safer place" in cargo) and in my experience you get it back on the ramp on the way out like they do with strollers. This may have changed since the last time I did it, or not be consistent but I think it is the best chance one has for the guitar.
If they let you on with it, they put it in one of the upright closets they have at bulkheads. When it won't fit in the overhead.
I have checked in guitars in soft shell cases, but it was always my "beater" guitar that I wasn't worried about. Always has gone ok. On the other hand, years ago in Kastrup airport waiting to board, I was watching a baggage handler in the Tarmac that drove his cart like 10 feet or more for the conveyer belt, and seemed to be making a game of throwing baggage (one was a set of golf clubs) from ten feet away in a high arc, to try and get it on the belt...and he wasn't very good at that even. By the time I thought to take video he was done. At least a third of the bags didn't hit right and fell off to the Tarmac. I don't think this happens often, but we passenger ought to be ready to film when we see idiotic, thoughtless, damaging abuse like that, and publicize it, so they react.
Good luck!