You can read "most underappreciated" to mean anywhere from "could be considered the best" to "widely but inappropriately ignored" to "broke the mold and started something new".
This list is not at all definitive; nor do I stand prepared to die on any of these hills. It is my first approximations made and refined with just a bit of supportive research. I will absolutely reserve the right to amend or disclaim any of this as I have only mild confidence that it stands up to more in-depth scrutiny. All that said, I'm right on all this, and you know it.
Category - Best | Most Underappreciated
Romantic Comedy - When Harry Met Sally | The Family Stone
Black Comedy - The Big Lebowski | Beetlejuice
Mockumentary - This Is Spinal Tap | Best in Show
Buddy Story - The Blues Brothers | Swingers
Christmas - National Lampoon's Christmas Vacation | Elf
Time Travel - (tie) Groundhog Day and The Terminator | Midnight in Paris
Sports - (tie) The Bad News Bears and Rocky | Caddyshack
Prison - Shawshank Redemption | Stir Crazy
Cop/Detective - The Silence of the Lambs | Fletch
Mafia - The Godfather: Part II | The Godfather Coda: The Death of Michael Corleone (aka, The Godfather: Part III)
Law - A Few Good Men | True Believer
War - Saving Private Ryan | Braveheart
Love Story - Casablanca | 500 Days of Summer
Redemption - The Green Mile | My Blue Heaven
Star Wars - Rogue One | Revenge of the Sith
Thriller - The Departed | xxxx [originally The Dark Knight; subsequently moved below]
Suspense - Memento | In The Bedroom
Sci-Fi - The Matrix | Alien
Horror - Poltergeist | The Blair Witch Project
Asylum Setting/Mental Illness - One Flew Over The Cuckoo’s Nest | Crazy People
[Updated 7/3/22]:
Western - Tombstone | High Noon
Fantasy - The Lord of the Rings (the trilogy; yes, this is cheating) | The Princess Bride
Super Hero (men acting as gods) - (tie) The Dark Knight and Iron Man | Ghostbusters
Super Hero (gods acting as men) - Superman | The Incredibles