What would be sufficent evidence to prove that the super natural exist?

Re. miracles:

Every natural occurrence is a function of particles and the fundamental forces of nature. Quantum mechanics describes the behavior of particles at the most fundamental level. I'm no expert, but I know that we cannot determine or predict exactly what a particle will do. Theoretically, it could do anything. So we can never say that something is an absolute impossiblity. But we can predict that particles (and matter at the macroscopic level) will behave in certain ways to a very high order of probability. So maybe a miracle can be defined as a macroscopic event for which the only possible natural explanation would be an extraordinarily unlikely quantum variation.

Healing of illnesses is complicated by too many variables. A better candidate for a miracle would be something very simple. An example: A pan of hot water is placed in a refrigerator. And without any application of energy, the water spontaneously comes to a boil, while the air in the fridge becomes colder. The normal downhill flow of heat from a hotter to colder region, as predicted by the 2nd Law of Thermodynamics, has been reversed. Assuming no outside energy has been applied to the system, this result--while theoretically possible as some kind of quantum hiccup--is so improbable that I'd call it a miracle.

Faith Unites said:

I think the real problem falls in trying to understand what the supernatural actually is. Just because God does something that can be explained doesn't necessarily mean that there wasn't divine interaction. Many of the miracles in old testament can be explained with natural phenomena but the real point is the timing of such miracles. When you combine timing with one of these phenomena you really start to struggle with just logically explaining something away. There will will always be people that just chalk it up to chance, but at the end of the day probabilities will still only take you so far. Ironically, it starts to make more sense to just believe in the miracle.

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Well that just muddies the waters even more, now the supernatural contains things that can be explained otherwise, but "just makes more sense" to attribute to Gods.

You've defined the supernatural to be "ill know it when I see it".

We know people aren't really very good at being probably calculators soooooo....

The supernatural is thus reduced to any time a human can comfortably interpret any given circumstance in reality as supernatural.