Roger Corman is excellent producer , director and writer . He was mentor to a number of famous directors, producers and actors. He also revived a numbers actors careers. He's well regarded in Hollywood.
The Navigator: A Medieval Odyssey
The Quiet Earth
Donnie Darko (That's a cult film, right?) Witchfinder General (A 'western' set in Civil War England) The Wicker Man (Goes without saying, the original) Theatre of Blood
Zardoz is brilliant, so ya boo sucks to those that doubt The Vanishing (Goes without saying, the original (2)) The Player
Picnic at Hanging Rock
The Company of Wolves
The Masque of the Red Death (yep a Roger Corman one, with Vincent Price, very classy) Society
A Scanner Darkly
The Thing (Surely that's a cult classic?) The Witch (2015 if there's other films called that. That one.)
I can't pick the best, as it changes like British weather. Could probably come up with more....
The list of cult films has grown big enough over time to make it's own public library of movies.
The Dark Crystal, Tremors, The Little Shop Of Horrors. The Loved One, skirts around the edge of horror in an extremely humorous fashion.
Are scifi movies from the 50's, such as Them or The Thing From Another World, that did good at the box office when released considered to be cult films?
If a movie has to fail at the box office to become a cult film, perhaps the really good ones were good at the start but fell victim to lack of a good trailer or blurb, the same way good books can fail simply because they have a bad opening and take too long to present a suitably baited hook.
Night of the Comet 
and Repo Man 
Both have a slightly odd beat of humour and make more than the best of their limited budgets. On top of that Repo Man has a great soundtrack and NotC has some great one-liners ["Daddy would have got us Uzis"].
Odd that they are both 1984.
For me, a cult movie is one you keep coming back to even if you don't know why... [Why do I love Night of the Demon  quite so much, when it is a pretty awful film?]
I'd call Sound of Music a cult movie now.
It was a great commercial success in the 60s but in the last 10-20 years has become an event/participation film to go to.
Some films have cult status due to time, geography, or message. For example, I feel that Buster Keaton's The General (1926) is unknown and yet one of the best films I've ever seen. Like the other comedic films I've seen, I expected Keaton's physical comedy to be exaggerated beyond belief... but his movements are so subtle and understated that I found myself riveted. You've seen Danny Kaye in The Court Jester, Jackie Chan in The Drunken Master, and Jim Carrey in The Mask.... and those performances are hilarious and the physical comedy demanded everything from the actors, but Buster Keaton goes to the other extreme and is brilliant.
As for geography, Bajrangi Bhaijaan (2015) was a major hit in India (nominated for Best Film, Best Director, Best Actor) and yet is probably unknown in the U.S. The plot is about saving a little girl, personal redemption, and international brotherhood with songs... well, imagine Ron Howard directing Forrest Gump as Jean Valjean... and that's basically it. It pulls at your heartstrings incessantly... It's about forty minutes (159 min) longer than it needs to be (for me).
Courageous (2011) is a no hold barred challenge to men. It's made by the Kendrick brothers, so that means a Christian message... and much less of a budget than a Marvel Avengers movie. The film opens with Nathan's courageous car chase with a surprise ending... it moves on to Adam's hilarious confrontation with Javier about his the terms of his employment... and deals with Adam and his family's heartbreak.
In the right demographics, all three of these films are popular.
El Mariachi. I love Robert Rodriguez's "we ride this crazy bus 'till the wheels fall off" approach, and this movie was insane. Great action sequences, an intense, yet, hilarious premise of a traveling singer being mistaken for a hired hit-man, and just a delightfully absurd piece. One of the best movies I've seen, considering it had a budget smaller than my petty cash fund.
It's a gargantuan list for me. Here's a bunch, listed in no particular order. Due to the graphic nature, read the titles at your own risk.
Split Second (1992) - Heavy Metal (1981) - Frankenstein Meets the Space Monster (1965) - The Green Slime (1968)
It Came From Hollywood (1982) - Strange Brew (1983) - Kolchak: the Night Stalker - (1972) Zombie (1979) - The Car (1977)
Rocket Man (1997) - The Golden Voyage of Sinbad (1973) - Suspiria (1977) - Zeiram (1991) - Beast of Blood (1970)
Wizards (1977) - Godzilla vs the Smog Monster - (1971) - Killdozer (1974) - Attack of the Giant Leeches - (1959) - Head (1968)
Taste the Blood of Dracula (1970) - The Neverending Story (1984) - Horror Express (1972) - J-Men Forever (1979) - Super (2010)
The Lost World of Sinbad (1963) - The Dungeonmaster (1984) - Galaxy of Terror (1981) - C.H.U.D. (1984) - Super Monster (1980)
Grave of the Vampire (1972) - Beowulf (1999) - The Giant Gila Monster (1959) - Roadie (1980) - Blood Bath (1966)
Black Magic With Buddha (1983) - Kung Fu Zombie (1981) - Hunchback of the Morgue (1973) - The Dragon, the Hero (1979)
Daleks: Invasion Earth 2150 A.D. (1966) - Swamp Thing (1982) - The Three Stooges in Orbit (1962) - The Deadly Spawn (1983)
Endangered Species (1982) - Infra-Man (1975) - Death Proof (2007) - The Beastmaster (1982)
Carnival of Souls (1962) The Shooting (dir. Monte Hellman, 1966) The Three Crowns of the Sailor (dir. Raul Ruiz, 1983 -- don't know if it counts as a cult film, but it should) The Hired Hand (dir. Peter Fonda, 1971)
And Eraserhead and El Topo, naturally.