Just about everything the New Mutants do, from stealing the Blackbird to continuing a party during a lock-down and then breaking into the Danger Room can be seen as either incredibly stupid or insanely awesome and hilarious.
A team of combined X-Men and Brotherhood members struggle with a mystical giant spider until Gambit steals a double-decker bus, intentionally flips it at high speed, climbs out a window and rides the sliding bus almost all the way to its collision with the spider, then charges it up and makes it explode point blank. Sure, it didn't stop the spider, but it was still awesome.
He manages to turn a Ship Tease into a crowning moment in his first appearance. He catches Rogue off-guard and leaves her practically lovestruck while handing her a charged playing card. Rogue takes several seconds to come back to her senses and realize she's holding a bomb.
Cyclops has a moment when the team is fighting Juggernaut. At one point, when everything they can throw at Juggie is being shrugged off, power beams included, Cyke takes off his visor and opens his eyes fully, sending Juggernaut flying back and utterly destroying everything in sight (literally).
Juggernaut: You think that fancy visor's gonna stop me?! NOTHING stops me! I'M RAW POWER!!!
Cyclops: Yeah? You want it raw, tough guy? Then take it RAW! (removes his visor and uses his powers at full force)
Of course, Juggernaut, being, well, The Juggernaut, eventually gets to his feet and powers his way through the blast, one step at a time, which was also pretty awesome. Then Rogue stole his powers, bitch-slapped him, and threw him into the river, encased in ice by Bobby on the way down.
Hell, let's just say the entire episode "The Stuff of Heroes" is a crowning moment for all involved. In the middle of a public witch hunt for mutants, this team risks their lives to stop a rampaging psychopath, and the entire country (or the government at least) has to admit they misjudged them.
And earlier in that episode, Scott's speech to Logan and the following fight with the military. Logan tells them to fight the military, Scott says no, they get out of this without becoming monsters. Logan insists he sees this as Black and White, Scott essentially tells him No, Logan ditches them, Scott and the X-Men take out the military without harming a single one of them.
Don't forget Cyclops unmasking Mystique, who has been impersonating Xavier for at least two episodes in the end of season 2.
And again in the very next episode (the season 3 premiere), Cyclops leaves Mystique at Area 51, and threatens not to let her come with them unless she discloses the location of the missing Professor. This plan doesn't work, but the awesomness still drips off of it.
And yet another for Cyke: at one point in "Target X", he and Jean take over teaching the New Mutants' class, and fail pretty badly throughout most of the episode. Then, in the last couple minutes, he decides to show them just how awesome he can be and has Jean lift all of the New Mutants into the air, before he fires an optic blast so that it bounces off the walls several times whilst avoiding every student in the air before coming down to slice an apple perfectly in half. No matter how you feel about Evo!Cyclops, you can't deny it was awesome.
A double one for Scott and his brother, Alex. Fully 'evolved' by Magneto's device, they break free of the mind control, and as they're running to get onto the X-Jet, Asteroid M begins to fall apart. In free fall just like they were separated, they grab each other's hand, then proceed to use their powers to slow down their fall, melting the ice at their landing zone and slowing them down. Asteroid M is still exploding above them and the X-Jet can't do a fast pickup, so they proceed to obliterate the entire asteroid with a constant stream of energy blasts until the thing explodes into harmless pieces, though they burn out the upgrades they got in the process and return them to normal.
Rogue gets a few (though the series largely seems to have been written to this end):
In the episode "Mutant Crush", Rogue comes to rescue Jean from Blob, who had kidnapped her earlier in the episode, even though she wasn't even on the X-Men's side yet. Oh, and she steals both Blob's and Scott's powers to do it.
In "Turn of the Rogue", Rogue and Scott are on a field trip with the geology class, and are waiting out the blizzard in a series of caves. Mystique (who Scott believes is just the school principal) has infiltrated as one of the students and reveals who she really is to Scott, just before knocking him off the cliff they're all standing on. Rogue is understandably ticked, and in a moment of fury, pulls her glove off and grabs Mystique's arm to absorb her memories, holding on just long enough to knock her out in the process.
Rogue: NO MORE SECRETS!
In the same episode, Rogue pulls a Heel–Face Turn and uses Cyclops's optic blasts to knock Mystique (who was about to attack the unconscious Scott) off the bridge made of snow they are crouched on.
Just to prove the point about this series being written to give Rogue crowning moments, she gets another in "Power Surge", draining off enough of Jean's out-of-control power so Jean can regain her senses.
Next is "Self Possessed", where Rogue loses control of her powers and starts recalling every power and person she ever touched. She proceeds to beat most of the main team with their own powers. A similar event happens in "Dark Horizon", except she was under mind control.
Is that the one where Wolverine tries to talk her down, and she uses the combined powers of Storm, Madrox and Cyclops to optic blast him from 10 directions simultaneously? Awesome.
Finally, when everyone is trying and failing to beat Apocalypse and his Horsemen in the series finale, Rogue takes Leech's power and single-handedly wins the day.
That wasn't so much as single handedly, the entire team was basically trying to distract Apocalypse so Rogue could do just that.
"Cruise Control" was pretty much a 20-minutes long CMOA for Amara aka Magma, the local Shrinking Violet. This is a shy, sweet little girl who jumps into a raging volcano, uses her powers on it to try stopping an eruption... and gets it.
Even Spyke gets one, via the episode "Uprising". In which he drives off a bunch of mutant haters with flaming quills.
One particular scene stands out for not only how mature Spike is acting, but how in a lot of ways he is doing the right thing. He reacts to a storekeeper refusing to serve him by knocking his purchases to the ground, before calmly walking out. Before Spike can leave however the mutant haters target a woman and her son. He reacts by standing between the two, knocking down a wall and allowing the boy and his mother to escape, before picking up what he wanted to buy and dropping a handful of cash on the counter.
Magneto gets one when fighting Apocalypse. Drawing on all his power, he pulls everything metal in the vicinity- including military vehicles, weapons, and satellites and hurls them at his enemy. As Apocalypse shrugs this off with with barely a reaction, this is also a CMOA for him as well.
Colossus gets one when he starts to get fed up with Magneto. Magneto uses his powers to pin him to a wall, threatening to "crush him like a can", so he turns into a human to escape his control and proceeds to punch Magneto, screaming at him to fight like a man. Unfortunately, Gambit stops him from actually making good on this. Though to be fair, he wasn't as urgent about it as he could have been.
"Operation: Rebirth". Captain America vs. a German fighter plane. The plane loses. Hell, pretty much everything Cap does in the episode is awesome, even more so because the Super Soldier serum in this continuity is causing cellular degradation and he is already suffering pretty badly.
In Cajun Spice, Rogue manages to be the MVP of the episode despite Gambit being at her side all along, and manages to take out several bad guys and find out just what's going on, using her power repeatedly to gain knowledge but never to borrow superpowers - wielding the least-seen-in-comics usage of her powers much more effectively than any combat power.
Blind Alley was one for Scott and Jean. Mystique tries to get revenge for Scott locking her in Area 51 for orchestrating a situation that nearly killed him and the New Recruits. To do so, she lures Scott to Mexico, strands him without his glasses in the middle of a desert and leaves him to die. Instead, Scott fights his way past coyotes to get to civilization, fights off several thugs when he accidentally destroys part of a hospital, fights Mystique one on one without sight and only sporadically using his power. Lures her into neutral ground where she can't see either, and fights her Daredevil style, and sees past her attempt to impersonate Jean. Meanwhile, Jean detects something's up, when Xavier insists nothings up, she checks on Scott's brother and finds out the truth, flies to Scott and finds him despite Cerebro's inability to find exact locations and psychokinetically beats the crap out of Mystique, before using a force field to push her into a wall until it colapses.
"Day of Recovery", when Wolverine is strapped to a table and interrogated by the military, while the other mutants are in cells with plastic walls. Wolverine uses hand signals to tell Spyke to launch a spike through one of the wall's air holes, striking one of the arm restraints and allowing him to cut himself free and take on the military.
What he does and says later after all the captive mutants are free is made all the more awesome:
Which in turn makes X-23 around ten times more awesome. She escapes Hydra, breaks into S.H.I.E.L.D., then turns up at the Institute and with the exception of Logan defeats all the X-Men. Every. Single. One. Berzeker mentions during her attack that not even Logan could take them all out. She then kicks Logan's ass and sends him running for his life until he is finally able to reason with her.
Her second appearance had her trying to fight off Logan who was trying to protect her from Omega Red and Deadshot who were hired by HYDRA to capture her. She makes no bones this is so he won't be a target and after they are cornered and captured X-23 reveals it was her plan all along, to get inside HYDRA so she could kill them all, and make it look like she died in the attack so no one else would come after her. She succeeds, evading even Nick Fury who was after her.
The President in "Day of Recovery". Everyone's out mutant hunting and on a witch hunt. What's the president saying? This is a time for fact checking, not accusations. Sure, he's got a whole security squad in front, but that's probably the Secret Service being paranoid. He's still arguing against the course of action that most of America is going at. In X-Men Evolution, at least, the White House has a good man.
Likewise, Nick Fury helping the X-Men... and having a full squad of troopers on hand with guns trained on the Brotherhood. If Nick Fury wants you captured, you're captured.
In one episode, Boom Booms father gets into an argument with Professor X and grabs his shoulder threateningly. Wolverine pops his claws and tells him those hands would look great on his dresser. The guy quickly backs off.
'Grim Reminder' after regaining control Wolverine goes on a Roaring Rampage of Revenge against the guy who made him what he is (metal skeleton, Brainwashed and Crazy), had enough willpower to break through the brainwashing (once it was cranked up to 100 it hurt) and with some help from Kurt and Kitty got his revenge. It's implied the guy did not make out of the exploding lab. To wrap up this example of badassery:
Kitty helping Logan realise who he is even though she's up against a crazy guy with claws, the fact that she didn't back down even after his claws nearly skewered her. Girl's got guts.
Kurt evading Sabretooth and able to teleport to Kitty's location with Sabretooth about to maul him in mid air.
In one episode, Kurt's teleports are revealed to pass through a hell like dimension, so Wolverine comes along as backup during a slowed down teleport. When they are attacked by the demonic creatures that live there, Wolverine is clearly willing to take on all of hell itself to protect Kurt.
The fight between Storm and Mystique in "The Cauldron, Part 1" is probably the most character-defining moment Storm ever had in the entire series and one of the defining examples of Break the Haughty in Western Animation. In this fight, she effectively not only punishes Mystique for all her mischief and trickery in the entire first season, but forever secures her dominance over Mystique and mentally scars that Manipulative Bitch for life by unleashing all her African might in one big punch to her smarmy blue-skinned face.
Blob of all people gets one when he actually makes the Juggernaut stumble by forcefully planting himself in the guy's way. He later gets tossed aside, but that takes guts.
Hell, the entire teamup between the Brotherhood and X-Men against Juggernaut counts with epic teamwork to remove his helmet.
When Mystique comes back to the Brotherhood house in "The He X Factor", Boom Boom decides that she's had enough with the place now that they're actually doing evil things again. When Mystique tells Boom Boom to clear out of Mystique's room that she had been using, she does so and then leaves the house...
Tabitha: Room's all yours...
(An explosion blows out all of the windows in Mystique's room and lights it on fire)
Kitty phasing the Blackbird through a mountain, never having used her powers for anything even approaching that scale before.