The very first one in the whole movie and Ariel's Establishing Character Moment would probably be the shark scene in the beginning. Seeing Ariel outwitting a shark like that is badass and pretty much solidifies her as the first Disney Princess to be an Action Girl.
Eric does a LOT of stuff that no reasonable person would do, and comes off as a major badass because of it.
The final battle, especially Eric who shows off how totally badass he is. His dispatchment of UrsulaDEFINITELY qualifies.
Eric returning to a flaming, sinking ship to save his dog Max. And when he's trapped trying to escape, he throws him free, showing himself willing to make a Heroic Sacrifice for him.
Hell, Eric has a ton of them over the course of the first movie. He's basically a prototype version of Aladdin.
Eric is insanely brave enough to destroy a freaking gargantuan human/cephalopod monster by stabbing it in its chest. You should never mess with him or his girl.
Not to mention his opening move in the fight is to dive underwater and throw a harpoon at Ursula.
Scuttle, after being little more than the comic relief for the entire movie is the first one to discover that Vanessa is actually Ursula and is told and quickly attempts to stop the wedding between Vanessa and Eric. Let's just say — it does not end well for Ursula in the slightest as Scuttle recruits the entirety of the ocean to rescue Ariel. It works.
In that same scene, Eric's dog Max gets a Moment of Awesome (and some well deserved revenge!) by saving Scuttle from being strangled by Vanessa/Ursula by biting her in the ass!
Keep in mind that Scuttle also exposed Ursula's plan in the first place, and would also destroy the amulet to give back Ariel's voice and free Eric from her spell (allowing his own Awesome moment in defeating her). Beforehand, Ariel was all set to accept the same fate as her novel counterpart in heartbreak, convinced she had lost fairly but Scuttle set off all the cogs in exposing and destroying Ursula's plan. You have a seagull to thank for Disney's happier ending!
Ariel rescuing her Prince from drowning was awesome! Granted he did save his damsel in the end (albeit after Ursula became a giant and by extension way out of her league), but you got to admit, Ariel is like the first Disney Princess to ever RESCUE her love interest!
Related to that, do you realize how frigging strong she must be? She's a little slip of a girl, yet is able to grab him, then pull him up to the surface and to shore with one arm, propelling herself with the other. Apparently, that collection of hers includes barbells.
Even more impressive, Ariel saves Eric's life again during the climax. Ursula points the trident at Eric and tells Ariel to "say goodbye to [her] sweetheart!" Quickly, Ariel grabs Ursula, changing the trident's aim so that the blast misses Eric for Flotsam and Jetsam.
Ursula is quite possibly the only villain to put together a plan and have it work completely, in part due to the Unspoken Plan Guarantee. She suckered Ariel, traded her for Triton and took over the entire ocean.
The scene where Ariel is naked from the seashells down after her transformation is about as awesome as it gets. Bonus meta points for the team being able to sneak that in.
Ursula's transformation from the beautiful Vanessa to the monstrous sea-witch. Seriously, only Ursula can mix Squick with pure awesome.
Any time Triton uses his trident usually becomes a CMOA.
The level of attention the animators gave to Ariel's hair. Even 25 years later, it still holds up well.
Not just her hair, but closely watch her facial expressions and body language while she's mute. The animators paid so much attention that they made her just as, if not more, expressive than she is when she has her voice!
Ariel herself is awesome as she is the first Disney Princess that actually did something and showed determination to make her dream come true.
Driven home by Ursula's little remark after the "Kiss The Girl" song. Maybe "little tramp" isn't quite the appropriate term, but can we all agree Ariel's having Eric ready to give her true love's kiss after two whole days was awesome?
As scary as it was,the way Ursula transforms into a gigantic version of herself is pretty damn awesome by itself. Her Family-Unfriendly Death too.
Triton threatening Ursula with the trident once she captures his daughter...unfortunately, she actually wanted him to show up.
Still, judging by what Triton does to Ariel's grotto and Morgana, in the sequel, if she had pissed him off enough, imagine what he could do to Ursula.
Sebastian and Flounder taking on Flotsam and Jetsam to save Eric certainly counts.
Pat Carroll clearly had an absolute blast as Ursula - especially in "Poor Unfortunate Souls". Hell, she was falling well and truly off the deep end of all sanity. Don't forget... she had nothing but a microphone in front of her and managed that! No wonder this woman loves to come back and portray Ursula. Even to this day we're not sure if Pat Carroll was on something at the time or whether she actually is just that insane.
According to the commentary of Fantasia 2000, the people at Disney were worried about how the people of Denmark would receive the changed ending, even warning Queen Margrethe II about it beforehand. Her response to the new ending was along the lines of "He never knew how to write an ending anyways."
The "In Harmony" number from "The Evil Manta." Ariel defeats the Manta's plans and drives him away... with one song.
Especially awesome when you remember that singing is Ariel's most noteworthy talent.
Earlier, Flounder gets one by tricking the Manta into thinking his brainwashing is working, then dropping the act the moment Ariel catches up to him.
In the episode "Thingamajigger", the Evil Manta gets his hands on the eponymous item (actually a boot someone dropped into the sea) so he can threaten King Triton with it. (At the moment the boot was found, a stray cannonball from ships fighting on the surface fell and destroyed the rocks the boot happened to be pointing at - so they assumed the boot did it!) Ariel, however, manages to scare him away by pointing two "Thingamajiggers" at him like a pair of guns. BootsAkimbo!
What's even better about it is that the "Thingamajiggers" Ariel pointed at Manta belonged to her future husband Eric.
In "Whale of a tale", Ariel adopts a baby killer whale, Spot, who got lost by his family. And he is too innocent, scared and childish to go back to the wild ocean. However, when three sharks attack Ariel, who has been like a mother to him, Spot gets mad. And the way he beats the crap out these terrifying sharks, is just... beautiful.
The music in this episode is so good it makes all the others look lame.
Whereas Eric saves the day, it's quite different in the stage/Broadway musical. Without her magic shell that she was given by her father, Poseidon, Ursula will die without it. In the climax, Ariel smashes the shell, destroying Ursula AND saving the day!