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What would it even mean to have evidence of something supernatural? Wouldn't having evidence (an observation?) make it no longer supernatural (above the natural)?

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I guess that depends on your definition of the word "natural". But one of the notable characteristics of nature as it actually exists in our universe is that its behavior is governed by discernible laws with very low Kolmogorov complexity. This is what makes science possible. But the universe doesn't have to be that way. Harry-Potter-stye magic could exist in principle, and that would be different enough from nature-as-we-know-it to deserve a different label despite the fact that magic would, in such a world, be "natural".

I suggest checking out Dean Radin's Real Magic. I'm in the middle of it now, and it seems persuasive in the affirmative.

It tries to use the scientific method to objectively determine if "real" magic (supernatural, psi, etc) exists.

I'll give long odds against.

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