It turns out that people are very bad at determining what's really random and what's actually due to a cause. And it also turns out that humans automatically try to shape their interpretations of events to put the best possible light on them, even when they are not, in fact, for the best. A few examples:
It may even be a survival factor for humans to think this way. There has been some evidence that depressed individuals are actually more accurate in some kinds of self-assessment than 'normal' people ("depressive realism"), though that's been called into question lately.
And, moreover, people can be fooled by things that sound like explanations ("God did it") but that do not actually explain anything. This article is an excellent discussion of the problems involved.
And finally there's the problem that a lot of things that should be possible for a miraculous God to do somehow never happen - like say, regrowing the limb of an amputee.
Well, not that any study's been able to show, in any concrete, replicable manner. At least, intercessory prayer hasn't been shown to make any difference in what happens. Praying for someone else, who doesn't know they're being prayed for, doesn't seem to make any detectable difference.
Now, can the process of praying make a difference in the person doing the praying? Can being prayed for when you know it's happening affect you? Sure. But in that sense it's not different from meditation or the placebo effect - one's mind affecting one's mind and body. This doesn't require supernatural explanations.
Of course, some people do claim to 'hear' answers from God:
When I talk about "hearing" God tell me to do something, I almost never mean that I literally heard a specific, identifiable voice addressing me in the third person (that did happen once, but it's not something I regularly experience). Rather, it's more of a sense of my thoughts being gently, almost imperceptibly guided down a specific path. Ideas might come to mind that I had not considered before and may even be foreign to my usual way of thinking; or I might suddenly feel overwhelmed with perfect certainty that a certain option is the right one; or I might gain a completely fresh perspective on the situation that I'd never seen before; etc. The more I am able to clear my mind (which has taken a LOT of practice since I tend toward ADD and always have a racing mind), the more I am able to "hear" these inspirations.
Consider for a moment the phenomenon of blindsight. Sometimes people have damage to the parts of their brain responsible for vision. Their eyes still work, but they can't process the data from the eyes normally. For example, they might be blind in one eye, or in an area on the left side of their visual field, or the blindness might be complete and they 'see' nothing at all.
Except, quite often, processing is still happening at a subconscious level. Some parts of the brain are still able to 'see' something, but this awareness never makes it to the conscious level. Ask someone with blindsight to 'guess' if something in their blind spot is red, or moving, though... and they do significantly better than chance (in exactly the way intercessory prayer doesn't). They just get a 'feeling' that there's an apple there, or a thrown ball, even though they don't consciously perceive anything.
We know that the brain works like this a lot more than people usually want to admit. We often get 'hunches' or 'gut feelings'. Some part of us has come to a conclusion; we're not consciously aware of all the data that went into that conclusion, but some kind of 'passing notes' happens that leads our conscious minds in a particular direction.
Engineers get this a lot. I can't tell you how many times I've realized the solution to a technical problem only after stepping away from it and thinking about something else. Riding the bus home, or in the shower, or out on a walk with my kids. I don't think any gods are feeding me technical inspiration; nor do I attribute the insights I get regarding personal relationships or my own struggles as coming from 'outside'. 'Clearing your mind' is vital to allowing such insights to 'bubble up'. Prayer can certainly work to do that - but so can meditation, or like I said, just taking a break from thinking about a problem.
No, to establish that such messages are coming from a supernatural being, we'd need messages that humans simply cannot come up with. How about the prime factorization of an extremely large number, say maybe ten billion digits? (Note: the message doesn't have to be ten billion digits long - for example, we know that 19249*2^13018586+1 is prime, that's 3,918,990 digits long.) It's relatively easy to check such a number with modern computers, but it's Very Very Hard to find such numbers. Even if that didn't prove supernatural influence (maybe it's a communication from advanced aliens) it'd still be very strong evidence that something unusual was going on.
Another possibility would be the ephemerides for a few nearby solar systems. This is relatively easy to check with some observation time on telescopes, but really hard to figure out from scratch. (One of the few things the movie "K-Pax" got right was the scene where Kevin Spacey's character does this.)
Instead, the messages from prayers all seem to be the kinds of things that people (or people's subconscious minds) can come up with on their own. (And if they turn out to be wrong, they are retroactively declared to be one's own mistake, or a message from Satan, or something. Only the 'gut feelings' that turn out to be right get attributed to God.)
I haven't, not in the sense of something precluded by our understanding of natural law. Nor have I seen any good evidence that such things have happened. Oh, I was (and still am) in awe at the births of my children, but that's not the same thing as a 'supernatural' ocurrence. My wife and children are "fearfully and wonderfully made", true, but that does not mean I think they are supernatural.
Besides, the fact that something is unexplained or not understood is not evidence that it is forever inexplicable, as I discuss here. Perhaps something truly remarkable did happen to you, but that doesn't mean that it was something forever beyond human ken. I recall the story of some peasants in 18th-century France who saw a meteorite strike. They called out some philosophers to study this strange rock that fell from space, only to be told that they must have really seen a lightning strike, attracted by this metallic rock that had been buried there...
One of my cousins told me about how her young son had talked about dreaming about a lady he hadn't seen before, who said she was "fat", even though she was thin. My cousin thought he had met a ghost of her aunt Fat, and the boy's description matched hers well. Uncanny, no?
On the other hand, consider how many dreams people have, and how many potential matchups there might be. It would be truly odd if uncanny coincidences didn't happen, and relatively frequently. A million-to-one shot happens eight times a day in New York City.
Now, this explanation of the dream may be wrong. Perhaps I'm like a French natural philosopher unwilling to believe that rocks fall from the sky. I'm willing to entertain the possibility, but right now statistics makes a pretty good explanation - one consistent with the known fact that humans are very good at self-deception. Absent reliable evidence for psychic or supernatural powers (and despite a lot of motivated reasearch such evidence is still remarkably absent) I'm going to have to go with Occam's Razor.
A real miracle, something we don't yet know how to do, would certainly pique my interest. But even then, how could we tell the difference between a god and a powerful alien? We'd have to listen to and evaluate what it had to say. If someone came up to me and touched my forehead and said, "Heal!", and suddenly I had perfect vision without glasses... Well, I'd certainly be willing to hear their spiel. But I'd still want some coherent explanations of things before I'd accept what they had to say as, well, gospel.
I guess, in the end, I'm not asking for anything more than Saul of Tarsus got.