Usually the process is the other way around. Do not go looking for situations where to use design patterns, look for code that can be optimized. When you have code that you think is not structured correctly. try to find a design pattern that will solve the problem.
Design patterns are meant to help you solve structural problems, do not go design your application just to be able to use design patterns.
Learn them and slowly you'll be able to reconize and figure out when to use them. Start with something simple as the singleton pattern :)
if you want to create one instance of an object and just ONE. You use the singleton pattern. Let's say you're making a program with an options object. You don't want several of those, that would be silly. Singleton makes sure that there will never be more than one. Singleton pattern is simple, used a lot, and really effective.