Accidental Innuendo: Piggy: "Hello, Loooooong John." Kermit: "What? Him too?" Apparently Miss Piggy dated him (in addition to Flint) after Kermit abandoned her. Try not to think about the details.
Big-Lipped Alligator Moment: The epic, random, and completely plot-irrelevant "Cabin Fever" number. Hilariously lampshaded when Clueless Morgan brings it up later, and his cellmates think he hallucinated it.
Billy Bones collapses...followed by Rizzo saying "He died!? And this is supposed to be a kids' movie!".
Then there's Dead Tom who is just a skeleton being casually dropped after its stated he's always been dead. Then there's Blind Pew who keeps bumping into things because he's blind. But you laugh anyway because he's a jerk and is Large Ham in doing so.
Blind Pew: [While the cabin is burning down] I think I smell… Something burning, no?
Talking Vegetable: Well, how else were we gonna get him in this movie?
Rizzo's side business of Rat Cruises - well, rats were a common pest on ships.
Hilarious in Hindsight: Captain Flint: that terrifying, human pirate who shot his own men, who we spend most of the film thinking of as a dreaded memory? Turns out he dated Miss Piggy. Just think about that for a minute.
Magnificent Bastard: Long John Silver. Magnificence includes his convincing the gang of pirates to grovel at his feet, begging for his forgiveness, approximately thirty seconds after they have given him the Black Spot. While he is tied to a tree.
Captain Smollett spares a few minutes to pay his last respects to Mr. Arrow. The scene is rather poignant, mood whiplashes and the revelation that he's alive later in the movie aside.
Viewer Gender Confusion: They just had to find the most androgynous looking boy in the 90's, dress him up in frilly clothes, and give him falsetto songs, just to confuse the children. According to the commentary, the actor actually went through puberty during the filming, so they had to over-dub his singing performances with older recordings of the songs.
This was lampshaded by Blind Pew early on, who kept calling/refering to him as a girl.
What an Idiot: They don't call the guy Clueless Morgan for nothing.
Jim had one moment where he told Long John Silver where the map was. Within earshot of Mad Monty, Polly, and Clueless Morgan. Say it with me now.