I'm not going to rule out the possibility of something existing that we don't know of yet.
However, if we're supposed to look for anything in any way, we first need to know what we're looking for. So, what's a ghost? Is a a soul, and if so, what's a soul?
I think that's an interesting question with or without ghost.
People who believe in souls, should attempt to explain what a soul is and how they experience their existence. Is it an emergent feature of the electrons or brainwaves that can travel in other dimensions that our normal physics can't detect or something completely unknown?
People who have experiences with ghosts and souls should be investigated. How do they detect their observations? Are they somehow able to sense things that happen in other dimensions? That'd be really cool, and I still won't rule it out. The human mind can do weird stuff. For instance echo localization like bats do, seeing more colours than normal or even just perfect pitch. Stuff like that is provable and shows that our senses can be expanded by training. Perhaps we even have dormant senses like seeing magnetic fields like birds do.
So let's say some people can see ghosts but we just can't measure it currently, because we don't know how it works. We can still make an experiment where we compare the observations from these special skills and see if they align. If it turns out that the people who can see ghosts agree, then we should definitely investigate it further and find out what happens in their heads when they see ghosts. Where does the brain impulses come from? That'll teach us about the special sense, which could then prove the existence of ghosts.
I doubt their observations match up in a controlled environment. It's a shame because it would be really easy to prove the existence of ghosts.