Musicals:
- Singin’ in the Rain
- West Side Story
- Top Hat
- Moulin Rouge!
- Cabaret
Musicals:
Romance:
War Movies (not all about the combat):
That last is actually an episode of the excellent TV series Foyle’s War, but hey, it’s movie-length.
If I must stick with theatrically-released films, though, then I’ll substitute The Land Girls.
Parodies:
Didn’t want to repeat any directors or styles, otherwise it would just be Brooks and Zucker.
Noir:
The Postman Always Rings Twice
Double Indemnity
Blood Simple
The Killers
Bad Day at Black Rock
Insanity/schizophrenia/mental illness (people going crazy):
The Snake Pit
Repulsion
The Three Faces of Eve
Sybil
Gaslight
CornTheHead, it was The Searchers and not The Seekers for the list of Westerns in the OP.
Heist movies
The Italian Job (original version)
Ocean’s 11 (remake version)
The Thomas Crown Affair (remake)
The Usual Suspects
Snatch