As a Christian, it never bothered me that much that there wasn't evidence of anything spiritual. When anyone would challenge me and and ask for evidence of a soul, angels demons, or even God himself. I'd argue back that all those things exist in spirit, not in the physical world. So, of course, you can't have physical evidence of something spiritual. The problem of this line of reasoning is that it leaves anything "spirit" in a category of being unable to effect the physical world. If that is true, if anything spirit is unable to effect the physical, than it's equivalent to not being there at all. For example, imagine a creature that exists only as spirit. This creature cannot affect the physical world in any way. For the sake of argument this creature truly does exist. What could this creature do or what experiment could be performed to demonstrate that this creature exists? In the end, if there are no ways to detect this creature in any way, then from our perspective as living in a physical world there is no discernible difference between that creature existing or not existing.
But, that's not the claim of Christianity. Not only is the claim that spiritual beings exist, but that they protect you (as in guarding angles), they can attack you (as in demons), they communicate with you, and ultimately guide you through your life. The claim really is that not only do these spirit being exists, but that they control what happens in the physical world everyday. If this is true, then we should be able to collect data that demonstrates, at the very least, that there are forces at work that can not be accounted for by ordinary means.
In astro-physics right now there is a determined effort to explain a force that for now is simply being called "dark energy". What we can determine is that the gravitational forces that we are aware of are unable to explain the effects of distant galaxies What's even more puzzling is that this unknown force has a greater effect on these galaxies than the forces that we can account for. We don't have to be able to explain fully what's causing these effects to realize that something is happening that we can't account for and needs further investigation. It's something like this, that I believe should be present in everyday circumstances if spirit forces are at work. I'm not saying it would conclusively show that supernatural is present, but it would give credibility to the supernatural claims.
One effect that I am aware of that has been scientifically investigated are those of prayer. A list of these prayer studies are at Wikipedia:
At the end of the article there is a final analysis. The analysis of all the prayer studies seem to be that either there was no effect, or that the effect was so small that it could be accounted for due to errors in collecting data or poor study controls.
A counter argument that has been given to me is that, there are studies that do show an effect.
These studies showed that the persons religion (or even lack of religion) was inconsequentual. What was truly being tested was a form of meditation, and the positive result was what you would expect from placebo, or having a positive attitude. Certainly God should be able to perform better than placebo, and that still leaves the best objective scientific studies showing no positive results. Objective evidence is the only tool that we have to determine true causes, as opposed to someone's personal belief of what is causing something to happen.
I think it's possible to perform scientific tests to look for spiritual effects. And, I think that if there is anything spiritual out there, we should be able find ways in which we can detect those effects.