Simple: create a single instance of your file manager (or whatever) class, and then pass it around inside your application as necessary. Much will depend on your application structure, but typically you have some sort of controller object that creates the other important objects in the application. That might be the object to instantiate the file manager, and then pass it to the other objects it creates that need a file manager.
If you're using a singleton because there MUST be no more than one instance of a certain class, that's usually okay. If you're using it because a singleton is a globally-accessible object that lets you avoid thinking about how the other objects talk to each other and what each object is responsible for, that's where you start to run into problems.
The file manager is a nice example. At first it seems like there should only ever be zero or one instances of the file manager object. But is that necessary? Can't you have two file systems on one machine at the same time?
