We can experimentally verify spirits by communicating with them. Many such cases.

Now, do you mean measure them using our physical devices that we currently have? No, we can't do that. They are "minds beyond ours" as OP suggests, just not in the way that OP assumes.

Djinn: Demons. Santa Claus: Saint (i.e. soul of a righteous human). Demons: Demons. Fairies (real, not fairy-tale): Demons. Most spirits that you're going to run across as presenting themselves involuntarily to people are demons because demons are the ones who cause mischief. Angels don't draw attention to themselves.