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
One of the ways you use a singleton is to cover an instance where there must be a single "broker" controlling access to a resource. Singletons are good in loggers because they broker access to, say, a file, which can only be written to exclusively. For something like logging, they provide a way of abstracting away the writes to something like a log file -- you could wrap a caching mechanism to your singleton, etc...
Also think of a situation where you have an application with many windows/threads/etc, but which needs a single point of communication. I once used one to control jobs that I wanted my application to launch. The singleton was responsible for serializing the jobs and displaying their status to any other part of the program which was interested. In this sort of scenario, you can look at a singleton as being sort of like a "server" class running inside your application... HTH