When should I use a private constructor in my class?
When you want a constructor, but don't want to expose it to the world. This could be because you have a factory method that calls the constructor (after validation), or because that constructor is called by ctor-chaining (i.e. public Foo(string) : this() { ...}
).
Additionally, note that reflection code is often able to use a private constructor - for example serialization or ORM libraries.
Also, in early C# compilers, when you are writing what would now be a static
class - having a private constructor was the only way of making it appear uncreatable.
When should I use a static constructor in my class?
When you need to initialize some static state prior to that state being accessed by instances or static methods.