Shared (or Static) Methods versus Singleton Methods

I was in the middle of refactoring an entire class library this morning when the solution to a few of my problems came in the form of changing a Shared method to a Singleton method, and vice versa. The question then hit me, “Is there something I don’t know about the tradeoffs of using each?” I set out to research this a bit, reminding myself of some things I have forgotten over time.