Cool School

This is the CoolCar and its vehicular bretheren's educational equivalent. It's often an AcademyOfAdventure and/or an ElaborateUniversityHigh, and frequently an ExtranormalInstitute, 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 ExtranormalInstitutes often fall into this, including but not limited to those that give degrees in...
 * Villain Ball Scholarships
 * Gothic Social Studies
 * Freelance Adventuring
 * Sneaky Zen Masters Degree (for being one with the Life Energy)
 * Fantasy Empire Leadership (when a tutor would do just as good a job)
 * Spy School Becoming Licensed to Kill people with Volcano Lairs on an island in the middle of the Pacific Ocean (or is in the process of building one)
 * Metahuman First Response Training
 * Magical Doctorates

...unless they're the kind to avoid like the plague.

Anime and Manga

 * Train + Train takes place aboard a Cool Train High School. Needs More Love.

Literature

 * 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.

Video Games

 * Psychonauts has Whispering Rock Summer Camp, where future Psychic Espionage Agents learn how to enter people's minds.

Other Media

 * |The Bottle of Ruin Academy for Young Pirates is too cool to be a fictional school, instead just a single image of a class schedule.