Cliché Storm: A force five hurricane, especially the death scenes.
Crosses the Line Twice: The time bomb that announces it's a time bomb and reads off its own countdown is hilarious. Even Rico winces over it, as he's dropping it. It is described as a terror weapon. It isn't intended to kill the enemy, its purpose is to scare the shit out of them.
Klendathu Drop, played several times in the film during battle scenes and over the end credits.
It's since become a staple of other movie previews and some award shows.
From the sequel, Hero Of The Federation, we have "Dax's Last Stand". It doesn't compare much at all to "Klendathu Drop", but it's still quite awesome as used in the film. Hey General!
Harsher in Hindsight: The nickname for the alien race that T'Phai belongs to, "skinnies", was originally from the novel. It was later used as a term denoting enemy combatants in the Mogadishu conflict of the '90s.
Misaimed Fandom: The movie is decidedly antiwar, portraying the humans as an fascist regime committing genocide on an alien race demonised through propaganda...and yet, it's one of the predominent Trope Codifiers of heroic Space Marines in science fiction played completely straight.
Recycled Script: This film was originally to be an Aliens ripoff called "Bug Hunt", then Verhoeven got the rights to the Heinlein story. Many wept that day.
Rooting for the Empire: A lot of people were rooting for the Bugs. In the first movie this might have been the filmmakers' intention, but in the sequels the Federation were supposed to be the good guys and audiences still found a bunch of giant cockroaches to be more sympathetic.
Spiritual Licensee: Aliens is often considered to be the most faithful adaptation of Starship Troopers ever made, with plenty of similarities between them.
Halo owes a lot to ST, especially with the armor and ODST's drop pods.
StarCraft has plenty of references to and points of inspiration from ST as well.
Space Marines in powered armour fighting large bugs? Warhammer 40,000
What the Hell, Casting Agency?: Casper Van Dien, Denise Richards, and Neil Patrick Harris playing Argentinean high schoolers... Verhoeven aimed for a Do Not Do This Cool Thing-baiting feel with the casting, going for stupidly beautiful people who would be far more at home in a soap opera—and then not telling them the film was a satire, leaving them to play their roles deadly serious.