When I create an application, I typically create a static class that contains static methods and properties that I can't figure out where to put anywhere else.
It's not an especially good design, but that's sort of the point: it gives me a place to localize a whole class of design decisions that I haven't thought out yet. Generally as the application grows and is refined through refactoring, it becomes clearer where these methods and properties actually ought to reside. Mercifully, the state of refactoring tools is such that those changes are usually not exceptionally painful to make.
I've tried doing it the other way, but the other way is basically implementing an object model before I know enough about my application to design the object model properly. If I do that, I spend a fair amount of time and energy coming up with a mediocre solution that I have to revisit and rebuild from the ground up at some point in the future. Well, okay, if I know I'm going to be refactoring this code, how about I skip the step of designing and building the unnecessarily complicated classes that don't really work?
For instance, I've built an application that is being used by multiple customers. I figured out pretty early on that I needed to have a way of separating out methods that need to work differently for different customers. I built a static utility method that I could call at any point in the program where I needed to call a customized method, and stuck it in my static class.
This worked fine for months. But there came a point at which it was just beginning to look ugly. And so I decided to refactor it out into its own class. And as I went through my code looking at all the places where this method was being called, it became extremely clear that all of the customized methods really needed to be members of an abstract class, the customers' assemblies needed to contain a single derived class that implements all of the abstract methods, and then the program just needed to get the name of the assembly and the namespace out of its configuration and create an instance of the custom features class at startup. It was really simple for me to find all of the methods that had to be customized, since all I needed to do was find every place that my load-a-custom-feature method was being called. It took me the better part of an afternoon to go through the entire codebase and rationalize this design, and the end result is really flexible and robust and solves the right problem.
The thing is, when I first implemented that method (actually it was three or four interrelated methods), I recognized that it wasn't the right answer. But I didn't know enough to decide what the right answer was. So I went with the simplest wrong answer until the right answer became clear.