The glorified global variable - becomes a gloried global class. Some say breaking object-oriented design.
Give me scenarios, other than the good old logger where it
A singleton should be used when managing access to a resource which is shared by the entire application, and it would be destructive to potentially have multiple instances of the same class. Making sure that access to shared resources thread safe is one very good example of where this kind of pattern can be vital.
When using Singletons, you should make sure that you're not accidentally concealing dependencies. Ideally, the singletons (like most static variables in an application) be set up during the execution of your initialization code for the application (static void Main() for C# executables, static void main() for java executables) and then passed in to all other classes that are instantiated which require it. This helps you maintain testability.