I was looking at different kinds of musical degrees and started wondering what jobs would require one. I know teaching would, but what others are there that require a musical degree of some sort?
A degree in music is one of those that you pursue because you love the subject matter, not because you think it will get you a job. Odds are that a degree in music will only qualify you for jobs that ask for an unspecified college degree, and then you'll have to have something else that they want. Yes, there's music education, but there are far fewer jobs than there are degreed folks in the field.
The biggest field that degreed musicians go into (other then the fast food industry) is teaching, either at the local music shop or at a conservency. The next biggest industry is probably commercial music - jingle writing, background music in TV shows or TV or radio commercials, music used between breaks on TV and some radio shows, etc. You need to be a "jack of all trades, master of none." The TV show producer will say that they need some sort of music for a segment of a show, and then you're expected to produce 4-6 songs for them to pick from. For example, they probably don't have enough money to get a certain AC/DC song, so they'll ask you to write a "song like that 'Dirty Deeds' song." Next, they might ask you for an R+B ballad or a Mozart-esque classical piece for a different scene. You then make a half dozen songs for them to choose from, etc. Hard work, but rewarding if you can get it.
The question is not whether your first job requires a degree, but where you plan on going with your life and whether you will need one in the process. The answer is Yes. No music degrees I know of lead to jobs except a double degree in Education and Music...but that's not a reason to avoid music. Double major in Business and Music and you can go just about anywhere.