We've got the experts to help. I have seen singleton classes used in many places. The difference between using a singleton over a class with static methods boils down to effective object-oriented design.
Singletons normally represent a cleaner approach.
A class of static methods, unfortunately, breaks down to a simple list of functions, or utilities. With a list of functions you no longer perform object-oriented programming. In fact, your work easily devolves into a classic procedural program. Suddenly, objects no longer represent the focus of your program. You begin to fall into a data-centric programming mode.