This typically falls under the field of pseudo-science, although there have been rigorous, empirical attempts to validate the supernatural (all have failed.) However, these inquiries themselves have formed a type of modern mythology.
Franz Mesmer (1734-1815) was a doctor who used a form of scientific method in an attempt to confirm a form of natural energy that connects living beings. Initially called "animal magnetism", his theories led to what we today call hypnosis (aka "mesmerism").
Wilhelm Reich was a doctor who investigated the paranormal. He is most famous for the concept of "orgone", a type of cosmic/biological energy. (His ideas may have provided a basis for "the Force" of Star Wars fame.)
Project MKUltra was an actual CIA research project that explored the possibility of scientific means of mind control, which appears in many mythologies. This endeavor has been subsequently popularized and extended in popular media.
Remote Viewing, a means of seeing things in distant locations, and even in the past, was studied, initially within the Spiritualist movement, but later by the CIA, the British government, and and the PEAR project at Princeton. Although none of these studies were a success, an early 19th century researcher includes Michael Faraday, who made significant contributions to the field of electromagnetism & chemistry.
Remote Viewing was a subject of the hit television series, Stranger Things and the film The Men Who Stare at Goats.
- Isaac Newton, one of the most important & influential scientists in human history was an occult researcher.
As with Faraday, the boundaries of hard science were still being defined. The mystical art of alchemy likely provides some precursor to modern science & empiricism, chiefly chemistry, medicine & psychology. (A chief goal of alchemists was converting "lead into gold".) Similarly, numerology can be seen as a precursor to number theory and the modern predictive sciences, which utilize statistics (manipulation & analysis of numbers) to chart the path of planets and achieve reproducible results in all sciences. Before the term "scientists", those making inquiries into the sources of natural phenomena were often referred to as "Natural Philosophers".
Other examples can be found in the paranormal wiki. It also has to be mentioned that the debunkers of paranormal claims are often more famous than the promoters, chief among them Harry Houdini. In the contemporary era, this tradition has been carried on by the Amazing Randi and Penn & Teller.