What's the best way to remove a broken screw from my guitar?

If it's just the screw shaft left with no head, you can try using a hollow screw extractor to drill around it. These work well for stubborn screws, but be careful because they can snap easily if you hit the screw directly. It's a good idea to start drilling in reverse to create a groove first. Good luck!

You might want to tap the screw extractor into the hole with a hammer, then attach it to a drill to pull it out. Just keep in mind that it might not work every time, especially since your screw is really stuck! Alternatively, you could use a Dremel tool to cut a slot into the screw, which would allow you to use a flathead screwdriver to turn it. Give that a shot if the first method doesn't pan out!