This is the Cool Car and its vehicular bretheren's educational equivalent. It's often an Academy of Adventure and/or an Elaborate University High, and frequently an Extranormal Institute, but none of those tropes apply in all cases.
So then how do you know it's this trope? The Cool School is too cool to be your school, a school that's probably very good at teaching, thanks to no shortage of funding and/or a Reasonable Authority Figure (or a few). Expect the textbooks and the Applied Phlebotinum to be full of Dangerous Forbidden Techniques and Sufficiently Advanced Technology or Magic respectively if the setting is in any way fantastical. If it isn't, it's probably one of the weirder school types not in the Extranormal Institute list like the School of Seduction or Clown School and/or has a Wacky Homeroom if it's not a Wacky College. The Library will be at least well-stocked, and there's a chance it's a Great Big Library of Everything.
Despite this, it's not always sunshine and roses. The villain(s) will attack the school if there are any, the Rival School will almost certainly be full of spoiled rotten over-privileged Jerks With Hearts Of Jerk that put the Cool School's resident Jerk Jock and Alpha Bitch to shame (in the other way), and the homework is often brutal.
The Cool School is often a Boarding School but never an awful one (unless the villains have made a hostile takeover), and the protagonist will lament either often or on rare occasion about how much life at home sucks and they're much happier at the Cool School.
The Trope is often a Wish Fulfillment fantasy of the writer and almost always one of the fandom. Who wouldn't want to go to/have gone to this school? Even if it's dangerous, it's never the kind of place that treats you like National Student Database Entry #78956723567606 or just the $1,000,000,000,000 "donation" for a new school building, and what you learn there is going to be useful.
This is where you will find the Stranger in a Strange School, any schools which really are ordinary Boarding Schools but are in a City of Adventure, the normal Muggle public school that somehow manages to do its job well, and is the trope where you should list anything that doesn't fit under the normal examples of overlapping tropes but is definitely too awesome to be real.
The Extranormal Institutes, including those that teach something other than holding the Villain Ball, being a professional emo, saving a damsel, being a sneaky zen master who is one with the Life Energy, running a fantasy empire when a tutor would do just as good a job, defeating the bad guy with the Volcano Lair on an island in the middle of the Pacific Ocean or is in the process of building one, defeating the other supervillains everywhere else as well (or hiding from racists acting rashly out of fear), and of course magic will fall into this trope unless they're the kind to avoid like the plague.
- Train + Train takes place aboard a Cool Train School.
- As far as Pokémon trainer Ash Ketchum is concerned, the Pokémon School on Melemele Island is pretty damn cool, and so is Professor Kukui. It has to be to earn the approval of our Book Dumb Idiot Hero. It helps that it is a school for learning about Pokémon, and many of the lessons are hands-on.
- In Robin Tim's first girlfriend Ariana attends a magnet school for the performing and visual arts that she and her fellow students love. The school itself is never seen but she's quick to gush about it to Tim's public school friends.
- The Trope Codifier is of course Harry Potter's Hogwarts School of Witchcraft and Wizardry, the Wizarding School both in-universe and beyond.
- Ready Player One is more focused on the OASIS in general, though the protagonist, Wade Watts, starts out living in the stacks, a combination urban ghetto, third-world slum and trailer park, but attending school on the virtual planet Ludus. The planet has many carbon-copy school buildings and is as mundane as a school in virtual reality can get, though Ludus' schools do have a class dedicated to completing what is essentially the entire plot of the book.
- Wayside School is a thirty-story building, built sideways by an apologetic builder. It also has "no" Nineteenth Floor and the students and teachers are just as weird.
- 16-bit High School is a school for video game characters.
- RWBY and the spinoff RWBY Chibi have Beacon Academy, where they train Huntsmen to fight The Grimm. Much vague referencing of fairy tales while fighting like a badass with anime style and flair ensues, as does Cerebus Syndrome.
- Video Game High School takes place in a world where eSports are bigger than the FIFA World Cup, and the titular school teaches students to play video games at a professional level.
- In the Navy, sailors learn to sail aboard Training Ships. If you're thinking of writing about one and it isn't in space or otherwise not Like Reality Unless Noted, the real life "Ship Prefix" is... TS Name of Ship. Otherwise, feel free to use a Fantastic Ship Prefix.
- Most Universities that are well known for their prestige and/or zany student pranks.
- If you went to college to get away from home, there's probably a bit of this in your mind.