SOLID includes the Liskov substitution princicple which has the notion that “objects in a program should be replaceable with instances of their subtypes without altering the correctness of that program”.
Since static classes with static methods (a bit like the Math class) does not have instances at all, is my system considered SOLID if I have static classes with static methods?
LSP applies to passing an instance of a class into a method, having the method do some stuff with that instance, and often produce some sort of result. This doesn't matter for static classes since in C# you cannot create an instance of a static class.
Even more importantly, static classes are sealed and therefore cannot be inherited. This makes your question moot far as C# goes.
You could say that static classes are always LSP-compliant since you cannot ever produce a subclass that would violate that principle. You could also say that static classes are never LSP-compliant for the same reason.
final and hide its constructor. Once you do that, though, they behave similarly to C# - you cannot instantiate them or subclass them. You can declare an inner class as static, but that does not mean the same thing as it does in C#: it simply denotes a nested top-level class.
VB.NET behaves exactly the same way as C# in this case, far as I know.
Single responsibility principle: a static class easily follow this principle.
Open/closed principle: since static classes are sealed, they cannot ever follow this principle.
Liskov substitution principle: as above.
Interface segregation principle: doesn't apply to a single class, but splitting one large static class into smaller, more specialized ones could be a step towards following this principle.
Dependency inversion principle: static classes cannot implement interfaces, so any class using it will always depend on whatever implementation exists at the time. Static classes therefore violate this principle.
Since static classes do not satisfy all 5 criteria, they are not SOLID.
I would not classify such a class as object-orientated and therefore I would say that it cannot (and should not try to) meet the principals of object-orientated design.
These classes are merely a workaround to the inability to provide code outside of a class in languages like Java and C#. If they could be, they should be defined as standalone functions since they do not gain any benefits from object-orientation.
It's worth mentioning, since you specified no language, that in C++ you can pass classes around which have only static members and access those members via template, and therefore it is possible to substitute for a class with only static members, and arguably, the methods called form it's "interface".
