Singleton Properties

Ok, if I create a singleton class and expose the singleton object through a public static property...I understand that.

But my singleton class has other properties in it. Should those be static? Should those also be private?

I just want to be able to access all properties of my singleton class by doing this:

MySingletonClass.SingletonProperty.SomeProperty2

Where SingletonProperty returns me the single singleton instance. I guess my question is, how do you expose the other properties in the singleton class..make them private and then access them through your public singleton static property?

Or should all your other properties and methods of a singleton be public non-static?