After Ariel nearly gets the True Love's Kiss required to remain human forever, Ursula exclaims "That little tramp! She's better than I thought..." The second part of the sentence is said in a tone of grudging admiration.
At least the 1989 German version cranks the expression up a bit by translating "tramp" as "Schlampe", which is somewhere between being called a "slut" and being called a "bitch".
The "Under the Sea" musical number briefly features a "blackfish" that looks more like it came from a racist 1930s cartoon rather than an animated movie made in the late 1980s.
Ariel being pretty clearly completely naked from the waist down for at least five minutes of the movie after she gets her legs. Neither she or the others seems to think much of it, until Scuttle tells her that the first thing she needs to do is dress like a human, helps her put a rag over her body, then wolf whistles and tells her how great she looks. Now, imagine how that line would've sounded before she covered herself up...
It's a "blink and you'll miss it" thing, but when Ariel first springs up above the surface, there is a very quick glimpse of light that reveals her naked body as she throws her head back.
There's another blink-and-miss-it shot immediately before that. Shortly after getting her legs, there's a brief clip where you can see her bare ass while she's rising to the surface.
In the Encyclopedia of Disney Characters book, for the entry of this film, the book takes note of Ariel not wearing any clothes when she first becomes human, and identifies this as a "fun fact''. A wonder how they got away with that.
Scuttle asking Ariel "New seashells?" Referring to her Seashell Bra.
Ursula has an interesting way of telling Ariel about the "importance of body language" in getting a man's love.
In "Poor Unfortunate Souls": "They weren't kidding when they called me, well, a witch". Flotsam and Jetsam both cackle lightly right when she says this. Guess they got the joke.
After Ursula sings the line, "But on the whole I've been a saint, to those poor unfortunate souls!" She uses the plant she first wore as a shawl to vigorously rub her back, shaking her chest as the camera zooms in on it. "Saint" is not the word to be used, in more ways than one.
Milder example in the French version: to translate Ursula's line "They come floating to my cauldrons, crying “SPELLS, URSULA PLEASE”!", the translators put "Elles débarquent dans me chaudrons en brayant “URSULA SAUVE-NOUS”!". It literally means "They turn up in my cauldrons, yelling “URSULA, SAVE US”!", but both the words used for "turning up" and "yelling" are sexual slang terms that aren't very likely to be used in a Disney movie otherwise.
When Ursula is explaining the terms of the deal to Ariel, one of her tentacles casually curls around Ariel's right shoulder. Ariel, with a very uncomfortable look, pushes the tentacle away and watches it warily for a couple of seconds. It's easy to miss, but it makes the whole scene so much creepier.
While trying to get Eric to kiss her, right before he almost does, Ariel gives him a really suggestive look.
Although the song "Kiss the Girl" is of course about kissing, it does have the lyric "You want her! Look at her, you know you do!" Not "you want to kiss her," but "you want her."
The priest's knees happens to look as if he's getting an erection. According to Disney, nobody realized this one when they were animating the scene, but it's practically Fanon by now. The animation in question has been edited out of the Blu-Ray release.