Ah, well, in terms of prayer, I'm not especially part of the crowd that looks for miraculous proof, so I'm not even going to attempt to provide evidence in that regard.
Makes little odds anyway, as it's always going to be anecdotal anyway. Even examples that are medically documented can usually be dismissed by the skeptic as remissions (and they often are remissions, but remissions don't usually come for the asking, of course)
In terms of 'something' happening, well, if one is praying, then something is demonstrably happening. Maybe nothing measurable outside of the prayer, but there is the prayer itself. If prayer is said to be more for affecting the pray-er than anything else, then that changes the measurement somewhat.
The problem with addressing this issue in isolation is that, despite vague and misleading religious my own belief is not in 'the power of prayer', it's in 'the power of God' and so proving that a prayer has been answered involves all those other issues of sovereignty, time, faith, reasons for faith and so on. It's not something that a 'pat' answer can be given to, iyswim.