Wow, it really depends upon which theory of supernatural stuff we are talking about. First, we must answer the question of whether this supernatural stuff in any way affects the material world. If it does not, but is operating in some other plane having no influence in the material universe, then science cannot investigate it. But, that sort of non-interacting substance seems irrelevant. If on the other hand, this supernatural stuff interacts with the material world, then science can investigate the effects because we can measure events in the material world and hypothesize what might be going on behind the scenes.
If things are as Joseph taught that there is no such thing as immaterial matter, that all spirit is matter just of a finer type (this is called qualified monism), then science should be able to investigate it just fine. It would be no different than monitoring neutrinos, mesons and bosons, or identifying the spin on quarks. This is hard work, but completely within the realm of particle physics.
Even if you speculate that the "supernatural" stuff is some kind of energy, that is still no problem for science because energy is part of this natural, physical world. We can measure charge and electric fields, heck we can even convert energy to particles. Energy and matter are just different forms of the same thing, like ice and water. Remember E=mc^2. Science can investigate energy and has been for years.
What if supernatural stuff is a force like gravity, or the strong force? That would be completely within the realm of scientific investigation. That is not supernatural, but natural.
What if this supernatural stuff was completely non-material and really supernatural, above nature? Well, then we have a problem. How can a thing with no physical properties (no mass, charge, momentum, etc) influence physical objects? There has to be some kind of interface. A neuron fires when the charge inside passes a certain threshhold (-70 mAmps). It will not fire any other way. Now we can change the charge by pumping in charged particles or we can send a current through it. But, how is some non-natural stuff that does not have charge (it cannot or else it would be natural) going to get a neuron to fire when it otherwise normally would not? This non-natural stuff needs to be able to produce some natural property to be able to have any influence in this material world. And if it produces a natural property, then we got 'em as scientists, because we can measure that. We can detect charges just suddenly appearing out of nowhere without an identifiable physical cause.
So, if the supernatural interacts with the physical world, then we as scientists can study it. If it does not interact with the physical world, then it is irrelevant because it plays no role in what happens here. The reader might be excited by this point and say, hey, then let's start researching these supposed supernatural phenomena: astral projection, near-death experiences, esp, prophecy, miraculous healings, answered prayers, etc. Well, I hate to disappoint you, but scientists have already been researching these things. You can read their research and critique it or be persuaded by it. But, the net result is so far all of these phenomena appear to have very banal natural explanations - no cool mystical forces at work.
Now, you can doubt that research without every even looking at it if you wish and instead lean on your own anecdotal evidence. The mind is easily amazed. There is excellent reason to not trust ourselves over carefully controlled and peer-reviewed research. I wish all knew our current understanding of how the brain uses heuristics to come to wrong conclusions.
God, I am proud of science. I encourage all those who are skeptical about science, to investigate those concerns so that you might get rid of them, instead of holding on to undeserved prejudice.
Some might find the story of Dr. Susan Blackmore interesting. She is a scientist who started out as a believer in supernatural stuff. She was going to show those stuffy scholars that there really was something to astral projection, near-death experiences, esp, etc. She went in trying to demonstrate that they were just as they seemed, but held true to her rigorous training by designing experiments that were sound. After years or research she finally had to conclude that there is nothing mysterious about these phenomena.
If all things are natural, as it appears to be the case. Then, we do not have free will, but are determined. For, how can we make our neurons do something they would not normally do based on their chemical inputs. In my opinion, we evolved to have the illusion of free will.
A long time ago, Beenthere asked for clarification about my belief that we don't have free will. Well, here it is.
So, there you have it booboo, this is where I stand.