Think about it this way: you access static members via type name, like this:
MyStaticType.MyStaticMember();
Were you to inherit from that class, you would have to access it via the new type name:
MyNewType.MyStaticMember();
Thus, the new item bears no relationships to the original when used in code. There would be no way to take advantage of any inheritance relationship for things like polymorphism.
Perhaps you're thinking you just want to extend some of the items in the original class. In that case, there's nothing preventing you from just using a member of the original in an entirely new type.
Perhaps you want to add methods to an existing static type. You can do that already via extension methods.
Perhaps you want to be able to pass a static Type
to a function at runtime and call a method on that type, without knowing exactly what the method does. In that case, you can use an Interface.
So, in the end you don't really gain anything from inheriting static classes.