Most of the characters' first big appearances in the show:
Tails completely shuts down a secret military base, leaving scientists and soldiers alike literally powerless, clueless, and frantic, then blows a hole in the complex with a biplane, rescues Sonic and Cream, and flies them all to freedom like it was nothing, making his first appearance in style - and then clinches the deal by being so darn modest about it when Cream praises him for it. Even better, he was apparently planning on doing it all himself, but was pleasantly surprised to see Sonic was already on the job.
Sonic, facing down Missile Wrist in the first big fight of the series. In what is an extremely Establishing Character Moment, he doesn't even give Eggman the pleasure of seeing him intense, and instead goofs off and dances on the robot while it's powerless to stop him, then makes it destroy itself.
Eggman, for that matter, in what is not only his first scheme but is one of his most villainous moments, lays waste to Station Square just to make an impact on the world (no pun intended). It isn't even his casual destruction of the city's best defenses, or his callous crushing of building and person alike, it's the way he taunts the world as he does it, completely assured of his victory. At one point, when it becomes clear he can't just demand the Mayor give over the city, he very casually tells his robot to just destroy the city completely so they can leave and find some other way to take over the world. Luckily, that's when the Tornado shows up.
Knuckles doesn't get his for a few episodes after his first appearance, but "Sonic vs Knuckles" is one big establishment for him, both of his gullability and his badassery. At one point he demolishes a forest to take down Sonic, and when he finds out Eggman is lying, he climbs up his robot of the week fistfirst, leaps into the air, punches Eggman in the face (leaving him a Twinkle In The Sky) and comes down fistfirst, completely obliterating the robot. All in about five seconds.
Tails furiously flying up in the Tornado to have an epic air battle with Eggman's Egg Fort in his pajamas, because Eggman had the gall to wake him up while he was trying to sleep. The moment kind of fades quick, though.
Credit must be given to the Navy and Air Force in the two-part Attack on Eggman's Base episode. They're sent in on a Suicide Mission as the distraction force so that Rouge can infiltrate Eggman's tower with a demolitions team. Despite being told that their technology and ammunition are vastly inferior to Eggman's future-tech, they still go in all guns blazing - between the battleship and cruiser fleet and the fighter wing, they manage to surpass Eggman as the last word in the Macross Missile Massacre. And bonus points to Eggman - it does absolutely nothing while the robot attendants thrash the Armed Forces. Even allowing for that, a good number of the humans manage to survive, and can be seen regularly bailing out before being shot down.
Eggman and his henchmen get one of these in the Sonic X episode Teasing Time when they beat their way into a Metarex base in order to rescue Sonic. He then proceeds to reprimand Sonic for his angry Dark-Chaos-Fueled behaviour (which sounds awesome even in the 4Kids Entertainment dub). Sonic calms down. Actually, he looks a little guilty.
Even Eggman's usually useless henchmen get to show off their talents by restraining Black Narcissus before he can attack their creator. Admittedly, it's possible thast Black Narcissus was just going along with it.
While all of this is happening, Shadow is busy rescuing the very people Sonic came to save in the first place. Sonic (and the audience) doesn't even know about this until Eggman casually mentions that Shadow has already rescued his friends an there's no need for him to stick around any longer. Shadow gets a Moment of Awesome when he isn't even on the screen.
Eggman then turns right around and reveals that this was all a big display - he's not siding with the good guys and he only came to Sonic's rescue to prove just how dangerous he and his team can be (granted that was a shock for pretty much everyone), and to offer the Metarex a deal - he can either be their greatest ally, or their worst enemy, depending on how much they offer him. All of a sudden Eggman's past incompetence as a villain doesn't seem quite so bad.
Even the recently Rescued from the Scrappy Heap Christopher Thorndyke was also doing damn well at being awesome in this episode after breaking Tails' security codes, following Cosmo into the Metarex base, attempting to negotiate with the villain, realising that wasn't going to work and starting an escape plan, then very nearly succeeding and escaping with Cosmo, until the bad guy caught up with him anyway.
Chris gets another of these in the episode Shadow Has Returned (dubbed as Trick Sand) Seeing Sonic falling to his death, Chris pilots the X Tornado right into an onslaught of lasers. When Sonic tells him to go back, Chris ignores him and flies right into it in a blatantly suicidal move (it's one of those rare occasions where the dub makes it sound better with Chris's "I DON'T CARE!") This scene says a lot about how much the kid has grown up, asides for the fact that they both end up needing to be rescued by Shadow anyway, making this a Crowning Moment of Awesome for him, too. Chris's moments tend to end up subverted in this way.
Actually, he gets one which isn't subverted in the episode (dubbed title) Countdown to Chaos. The shock and anger of seeing Sonic kicked about like a football gives him the strength to break free from a set of solid metal restraints and then tear the Chaos Emeralds out of the still active robot with his bare hands (which for the record, really really hurts). Please note: this is a twelve year old child breaking out of solid restraints.This is a rare example where watching the dub may be better. If you're watching this scene in the original Japanese, the voice acting (or should that be screeching) makes it Narm.
Rouge punching out Knuckles during the episode Memories of the Wind.
Another Crowning Moment Of Awesome for Sonic came in the episode Fast Friends Sonic and Sam Speed race against each other to finally decide who's the fastest in Staion Square, with Sam racing along with a government designed Nasa jet engine attached to his car. The race appears to end in a photo-finish, until playback reveals that Sonic reached the finish line way ahead of Sam, hoped over the tape, leapt up to grin for the camera before returning to the race for the photo finish.
Amy Rose, in Memories of the Wind takes on Eggman's ship with about a million Piko Piko hammers (this is also the first sign we get that she can create as many of them as she needs and it's not just a single mallet stored in mallet space).
Cream during the second half of Sonic Battle, Face Off! takes on and defeats a beserk Emerl entirely on her own, comforts her Chao, Cheese afterwards, and then promptly bursts into tears when the aftermath of Emerl's destruction sends a shower of rain high into the air, creating a rainbow overhead. It's still awesome, not to mention a Tear Jerker.
Tails defeating Eggman (nearly) after Sonic's (supposed) death on Board Space Colony Ark. That's what'cha get for offing Tails' personal Obi Wan, Eggman.
Another one, in ep. 45 when he beats a kungfu guy in a fighting tournament. He's even able to put up an impressive fight against Rouge in the second round, after she initially believed he would be easy pickings. Makes you smile for the kid.
It's weird, but the Sonic Driver which fires Sonic as ammunition is undoubtedly one awesome machine. Particularly when they use it while in Supermode and with Shadow and Sonic at the same time (which was so powerful that it destroyed the Cannon itself).
Knuckles gets one when he defeats Yellow Zelkova in the Galaxtic Corridor - he actually smashed his own gloves/hands/whatever, he attacked so hard. Not to mention he then offered Zelkova a hand out of there, which he declined, choosing suicide over Knuckles' aid.)
Sonic gets another one in the episode Countdown to Chaos, after being half beaten to a pulp by Eggman's superpowered Chaos-driven Robot (which is holding Sonic's friend as a prisoner on board). Despite the fact that Sonic has been so freaking well beaten up that he's barely walking by that point, he still stands up, grinning, and approaches the machine telling Eggman that he's coming to get Chris back - and woe betide anyone who tries to stop him.
In the Archie Comics adaption of the anime, Dr. Eggman gets two of these while masquerading as the luchador El Gran Gordo. Initially a ploy to earn money, Eggman soon becomes addicted to the fame and praise of being a hero, despite using some mechanically-engineered cheating at first. Eventually, Sonic challenges him to a fight, which Eggman does lose, but still gets in a few good hits, all without the use of weapons or trickery. As the comic later confirms, the man himself lasted longer against Sonic than any of his machines did. More pertinent, however, is El Gran Gordo's return to the ring, where he faces off with Andes the Ginormous, a monstrous man roughly twice his size. The two go at it for awhile, but eventually, the Doc ends up exhausted and on the verge of tapping out, before he catches a glimpse of Chris in the audience sullenly saying "I believe in you", which instantly gives Eggman a confidence boost, complete with a triumphant shout of "EL GRAN GORDO NEVER SURRENDERS!", followed by him pummeling the ever loving heck out of Andes, finally knocking him out with the "Flying Gordita Smasher." He takes the championship belt, and wearily walks away, having proven himself to his public..
Let's all face it: Shadow's one powerful mother, right? The episode "The Cosmo Conspiracy", where Shadow tries to kill Cosmo because she's an unintentional spy for the Metarex. While trying to get Tails and Cosmo, Knuckles gets in front of them and not only manages to hold Shadow back, he manages to throw him into a wall. Granted he gets knocked out about five seconds afterwards, but anything is possible when you're VOICED BYDAN GREEN!!!
No respect for Tails taking on Shadow to protect Cosmo when all else failed? Sure, he was beat pretty bad, but give the kid credit for tenacity. Love inducedtenacity.
Hell yeah. he also managed to just evade Shadow using clever tricks with the computers for a good portion of the episode.
It's also a crowning moment of awesome for Shadow, as he singlehandedly takes out Sonic's entire group.
Even though he decided to go to Sonic's world in the first place staying there forever clearly wasn't on his agenda, so when Chris discovers that using the Master Emerald to help Sonic fight Dark Oak will probably prevent him from ever being able to return home; it's quite a big step for him when he tells them to do it anyway. And he tries to brush it off as no big deal, smiling even though he's clearly distressed. This is the kid who thirty episodes ago whined, clung, and was generally considered a royal pain by the fans through the whole series. Then he grew up. Yay for Character Development.
You have to give props to Maria. An idealistic twelve year old girl who had a gun pointing right at her back along with a clear threat to kill her if she so much as moved. But she still pulled the lever releasing Shadow and sending him to earth, thus sealing her own fate.
Super Sonic is guaranteed to end up in this page. Standing out in particular is the fight against Perfect Chaos, the first time we see the glorious golden hedgehog in action (aside from practically one-shotting a robot.)
Another moment that stands out is in episode 64, Metarex Melee, when Sonic and Shadow, when dashing after two Chaos Emeralds, end up activating the rest and transforming.
Episode 25, after Eggman tricks Knuckles and Chris into giving up six of the Chaos Emeralds, he declares that he doesn't care which planet he takes over as long as he fucking takes over! Hell yeah, baby! Makes you kinda wish season 3 never happened, doesn't it?
We also can't forget the scene where Bokkun is flying away in fear from a clearly pissed off Ella, the normally peaceful maid, driving the X Tornado, managing to transform it into Attack Mode, and kicking the living hell out of Bokkun. Needless to say, if you activate her Berserk Button, she can make anything work, even when normally she has no clue how to.
The absolute best moment in the original Japanese and French dub is during Super Sonic and Super Shadow's battle against Final Hazard and subsequent Chaos Control, all set to Live and Learn from Sonic Adventure 2.