Fantastic Science

Humans are naturally curious, and even when Science Is Wrong we want to know why. In a fantasy setting, these curious people will found whole new academic fields and professions around studying the strange and fantastic flora, fauna and phenomena that exist in it. Just like some schools of Hermetic and Vancian Magic will create Ritual Magic by analyzing occultism with scientific rigor, these professionals (not necessarily wizards, though that does help the survival rate) will go about cataloguing, studying and (hopefully humanely) experimenting with peculiar phyla in a very empirical way.

Possible Fantastic Scientists include a botanist who studies mandragora, a biologist cataloguing griffins, or an epidemonologist who studies outbreaks of infectious Demonic Possession to find a cure. Likewise, you can expect their field of study to have a strange made up name, like Herbology or Impology. All the same, they will go about studying these curious classes as if they were real... because in the setting, they are!

As a character, a Fantastic Scientist is often The Professor or at least The Smart Guy, possibly a bit of an Absent-Minded Professor or even Mad Scientist. They can be amiable enough deliverers of exposition, background, in need of rescuing from their subject of study, or the cause of some shenanigans (like the above epidemonologist letting loose an improved possession plague). Their motivations can range from curiosity, a desire to discover Potential Applications, or to gain kingship over this kingdom. Their home or office will usually have a Library of Babel with truly eclectic books and a Bazaar of the Bizarre composed of their subject of study.

A Magic-Powered Pseudoscience is more likely to be "alchemy works because of magic" and modeled on that pseudoscience than an attempt by in-universe characters to model the magic as the regular science that this is. It should be noted that alchemy does not fall under this trope due to the fact that it is technically real.

If an Occult Detective decides to also catalogue whatever it is they detect, they may also be researchers of a Fantastic Science. Research into magic itself is the subtrope Sufficiently Analyzed Magic. See also Magic Versus Science, Admiring the Abomination. Compare The Spark of Genius.

Examples of Fantastic Science include:

Anime

In Neon Genesis Evangelion, we have "metaphysical biology," which seems to be the science of souls. This fits with how souls are treated in the series.

Comics

Planetary
Obscure Wonder Woman enemy Doctor Poison calls herself a "Mythobiologist". She studies the biology of mythological beings and their effects on more mundane life.
Skybound Entertainment's comic Witch Doctor portrays the realm of supernatural creatures as parasites that are best battled by medical specialists rather then soldiers. The first issue of the mini-series treated a Demonic Possession as being similar to a parasitic infection, with the young victim, after being purged, having to remain inside his family home surrounded by protective runes until the main characters could find a more permanent cure for his condition.

Fan Works

Over the course of the various installments of Drunkard's Walk, Douglas Sangnoir (native of a version of Earth with Superheroes) makes occasional mention in both dialogue and narration of "metabiologists" in his home world who study the nature and function of superpowers; readers will learn some of their classification system and terminology, which powers are understood scientifically and which still are mysteries. And some of the worlds he visits have their own Fantastic Sciences.

Film

In Ghostbusters, two of the three original characters, Ray and Egon, are legitimate paranormal researchers that channel their knowledge into the business of "paranormal investigations and eliminations."
- It's not clear what Peter contributes to the research, if anything, though he was the one that came up with the idea of making it into a business after getting kicked out of Columbia University.
- Peter has a PhD in Parapsychology, so I always just assumed his part in the research was in that area.
Similarly, the 2016 remake has two legitimate paranormal researchers (Abby and Erin) and one paranormal engineer (Holtzmann) among the members of the "Conductors of the Metaphysical Examination". And unlike the originals they're not hunting ghosts for profit but to capture specimens they can analyze..

Literature

Harry Potter: Some of the Classes at Hogwarts operate this way, like Herbology. It's basically just Botany, but with magical plants.
- Infuriating because they seem never to study any basic non-magical skills. This is actually lampshaded by Hermione in the very first book, when she and Harry are confronted with Snape's riddle. She comments that an ordinary wizard may have had serious problems solving it, because they don't learn any math/logic puzzles.
Telemain of the Enchanted Forest Chronicles is a magician who studies the technical side of magic and has a habit of talking in incomprehensible Magi Babble, much to the chagrin of the other characters.
Theoretical magicians in Jonathan Strange & Mr Norrell study magic but don't ever practice it.
In David Eddings's Mallorean series, the Melcene University has schools devoted to alchemy, necromancy, etc.

Live Action TV

Helen Magnus in Sanctuary.

Helen: I specialize mainly in cryptozoology and xenobiology. Teratology, too, when the need arises.

- Those would be the studies of unknown animals, of aliens and of monsters.

Tabletop Games

Shadowrun had parabiologists and parazoologists. They studied Awakened (magical) plants and animals, respectively. It also had academics who studied the theory of magic but couldn't actually do magic themselves.
In GURPS, thaumatology is "the academic study of magical theory". Occultism has specialties like vampirology and pneumatology for more specific areas of study.
- Pneumatology is from the Greek (πνεύμα) and means "the study of spirits."
Artificers from the Dungeons & Dragons setting Eberron are described this way.
The Ravenloft setting's various sourcebooks on monsters, the Van Richten's Guides, are presented as the work of scientifically-minded researchers of the supernatural.
Exalted uses the term 'savant' to denote anyone who has a scientific understanding of the fantastic forces at work in and beyond Creation, as opposed to people who are just operating on superstition and folklore. Also, anyone who works extensively with Magitek is called a sorcerer-engineer.
- And it also uses the term "motonic" for a major field of high-tier scientific study, related to the way that everything in its world is composed of motes of Essence.
Eon plays it straighter than an arrow, since, in this universe, magic is science!
In Cthulhu Tech, sorcery is something you can learn in college. You might not actually want to though ...

Video Games

Pokémon professors. Professor Oak studies Pokémon-human behavior and interactions, Elm does Pokémon breeding, Birch studies Pokémon ecology, Rowan studies Pokémon evolution, and Juniper studies Pokémon origins.
The Tales (series) is fond of this, along with Sufficiently Analyzed Magic:
- Tales of Symphonia has Raine Sage, who is a magical historian with a focus on healing arts and magitechnology.
- Tales of the Abyss has Colonel Jade Curtiss, who (along with his former friend Saphir) used to be a sort of magical geneticist, and goes back to it after the game. Saphir Neis and Guy Cecil both focus on fontech, although Guy is really more of an enthusiastic hobbyist.

Web Comics

In Gunnerkrigg Court, the eponymous Court studies the extranormal from time to time. Kat Donlan discovered that the Shadow Men actually have mass, and the Suttons are botanists/gardeners who restore magically-animated trees to their original state.
Adventurers! gives us powerfologists. See Bishonen Line for more.
Tedd of El Goonish Shive seems to be heading in this direction.

Western Animation

From Foster's Home for Imaginary Friends, a couple of nerds were trying to start the scientific field of Figmentology, since imaginary friends are real on that show.
In Wakfu, the lost tech of the Eliatropes is described as "magical science". Nox, the Big Bad of season 1, is also such a fantastic scientist, studying the principles behind the wakfu and time magic to attempt Time Travel. Even before finding the Eliacube, as a humble clockmaker he was able to built a flying pocket watch—which just lacked a durable power source.
Dr. Thaddeus Venture, among other characters on The Venture Brothers, is a self-classified "super scientist", despite never having finished his degree.