Revealing mistake: Ariel swims underwater and opens a book with stiff and easy-to-flip pages that produce a flutter sound when turned. The paper should be either soggy, or with its pages stuck together or even dissolved, and in no case sound and look as if it were dry.
Continuity mistake: When Scuttle attacks Vanessa, she scares him away using her left arm. An instant cut later, both of her arms are raised.
Other mistake: When Eric holds up the telescope, the cap is on the lens.
Continuity mistake: Vanessa approaches Eric and admires his ring while they stand side by side. A shot later, he has turned 90º and is facing her.
Trivia: Jodi Benson, who voiced Ariel in the 1989 animated movie "The Little Mermaid", appears in the live-action film as a marketplace vendor who offers Ariel a plate of food and a fork. In a reference to the original film, Ariel uses the fork to comb her hair. (01:20:30)
Answer: Her long hair covered her up.
There are a couple of shots after her transformation where you can tell she's no longer wearing her bra. However, I rewatched her transformation scene, and her bra is never seen coming off. It's just gone.