It makes sense when you have one instance of some kind of data that makes sense to manipulate as an object. There is hardly anything you can't do with class variables and methods, or for that matter with global variables and normal subs - but the singleton pattern can make for more logical, easier to write and maintain code.
in reply to Singleton vs Static
You might want to initialize some data the first time it is accessed if it is accessed, that is done automatically whenever you try to get the singleton the first time. You might want to access the data in different ways, including getting it via calcualting accessors, that has no direct mapping to the data, etc. Many such cases makes more sense to wrap up in an object, although you could do this with any kind of global data instead.
It is a design pattern, and thus mainly a way to design your code in a hopefully better way for certain situations. It doesn't make something otherwise impossible possible. :)
You have moved into a dark place.
It is pitch black. You are likely to be eaten by a grue.