Continuity mistake: When Sebastian is hiding from the chef under a lettuce leaf in the kitchen, at first he is behind a bunch of stuff on the table. In the next shot, all the stuff is gone, and he's alone on the table.
Continuity mistake: When Flounder swims through an anchor at the start the anchor has round edges, but when the shark bursts in the anchor has flat edges.
Factual error: At the end of the film, when Ursula has Triton's powers, she creates an enormous whirlpool that pulls wrecked ships off the ocean's floor. Prince Eric boards one of the ships that is being swept along with the whirlpool's currents, then turns the rudder hard to steer the ship into Ursula. Not possible - in order for the rudder to change the course of a ship the vessel must be moving faster than the current.
Factual error: After Ariel is changed into a human, she swims frantically to the surface for air, and when she comes up she does the magnificent throwing her head back and splashing water everywhere - we've all seen it. But this would mean she would have to actually tilt her head further down in the water before breaking the surface, so she could then throw it back- something that makes no sense for a person who's about to drown, and wants air as soon as possible. Can't have been a "hair in face" problem - her hair was streaming out behind her underwater, so wouldn't have been in the way if she'd broken the surface face first.
Continuity mistake: When Ariel and Flounder are going into the shipwreck at the start, Ariel goes through the porthole and Flounder follows. In the frame where he gets stuck, his front fins are trapped firmly outside the porthole but in the next shot inside the boat, they appear inside instead of outside.