Without a good definition of the supernatural it's going to be awfully difficult to prove any cases. In fact, I just say I believe in God et al., but depending on how you define the supernatural, I am not sure I believe that supernatural entities exist. I can imagine saying that God, angels, and souls all exist, but that science just hasn't developed enough to analyze and predict the activities of these entities. So they are supernatural from the standpoint of present science, but then so are lots of things that science will someday discover. But since we don't know what "ultimate completed future science" will include we can't say for sure whether these entities are natural or supernatural.
If science has to screen certain entities out because they are by definition beyond the competence of science to analyze, then it is a boring result that science hasn't found evidence for them. Science, on that view, can't either support of deny their existence. On the other hand, if it is within the competence of science to show that these entities do not exist (as in Why the Evidence of Evolution Reveals a World Without Design), then if someone has offers evidence for something like God, then it can't be thrown out of court a la Judge Jones because science has to stick to the natural and not the supernatural.
As the Statler Brothers say, you can't have your Kate and Edith too.