I'm not a great OO theorist, but from what I know, I think the only OO feature that static classes lack compared to Singletons is polymorphism.
But if you don't need it, with a static class you can of course have inheritance ( not sure about interface implementation ) and data and function encapsulation.
The comment of Morendil, "The design style embodied in a static class is purely procedural" I may be wrong, but I disagree.
In static methods you can access static members, which would be exactly the same as singleton methods accessing their single instance members.
I'm actually thinking now that another difference is that a Static class is instantiated at program start* and lives throughout the whole life span of the program, while a singleton is explicitly instantiated at some point and can be destroyed also.
* or it may be instantiated at first use, depending on the language, I think.