Awesome Music: The Tangerine Dream score. Especially the...oh, to hell with it, the whole thing. If righteousness and beauty had a transcription in music, for instance, it would be the unicorn theme. The original score by Jerry Goldsmith is pretty awesome too. Because, well, it's Jerry Goldsmith.
Even so, Jerry Goldsmith was so upset by Scott's rejection of his score that he never spoke to the director ever since and up to his passing.
Angel/Devil Shipping: As mentioned below, Lili and Darkness is a popular pairing. He is implied to be the son of Satan and he is attracted to her because her soul is pure. Just contrast his appearance as the literal devil and she as an angelic young woman. Unsurprisingly, Foe Yay ensues.
Big-Lipped Alligator Moment: Well, there's plenty of weirdness on display throughout the film, but during Lili's dance with the animate black dress, an evil-looking cherub comes to life on the tabletop. Lili doesn't notice it, it doesn't intervene in any way, and once the sequence is done, it's never brought up again.
Broken Base: Over the score. Even those who like Jerry Goldsmith's score wish the American release music was retained for the Dress Dance.
Draco in Leather Pants: Darkness. Despite being a monster with cloven hoofs and a goat's face, his love for Lily is genuine, if twisted, leading to this trope. Being played by Tim Curry is also a reason.
Evil Is Sexy: Dark!Lili and Tim Curry as a shirtless, ripped, deep-voiced devil.
Fan Preferred Pairing: Lili/Darkness is actually quite popular. You can find a lot of fan fiction dedicated to this pairing.
Fridge Horror: In the U.S. Re-Cut, it's implied that Jack and Lily have sex. Which is fine until you remember the goblins are following them and tracking their every move.
The mere fact that one of the characters is namedGump.
The entire film is basically The Legend of Zelda before it even existed: a young boy goes on a quest to save a princess and prevent a demon king from obtaining/destroying a sacred object and conquering the world.
Mis-blamed: The film has a very true connection to Brazil; both are big-budget fantasy films from visionary directors, from the same producer (Arnon Milchan) for the same studio (Universal), released in the same year (1985), and both films were subject to scrutiny from the same executive (Sid Sheinberg) who demanded the director to make changes to the film before its release for commercial reasons. Because of this, fans blamed Sheinberg for the decision to replaceJerry Goldsmith's score with Tangerine Dream's for the American release, which Ridley Scott denies. Scott later admitted that the music change was his responsibility, because he was anxious after failed test screenings, and this stress led him to confide in Sheinberg, for whom he firmly believed was helping him save the film. It didn't help that Blade Runner was not a box office success when it was first released, which probably made Scott very nervous about being unemployed.