The fight with Storm Eagle in Mega Man X1 deserves a special mention. Before the fight begins, X hitches a ride on Eagle's Cool Airship, the Death Rogumer, which starts to rapidly ascend. X blasts off the forward guns, climbs underneath the ship and sneaks into the ship itself. A few seconds later, the top half of the airship explodes, prompting Storm Eagle to arrive. When you defeat Eagle, the crippled ship starts falling down towards the Power Plant Stage.
Mega Man: Maverick Hunter X dispenses with X crippling the airship prior to the boss fight, but makes up for it with Storm Eagle having one of the more awesome boss banters in the game.
Zero's Big Damn Heroes moment in the intro stage of X1. Vile has curbstomped X in a Hopeless Boss Fight and has him in the death grip of his Powered Armor. After Vile taunts X, we hear the sound of an X-Buster charging, and then a charged shot blasts off the arm holding X. Cue Zero racing in and saving X's metal hide.
Taking on the boss redux fights with the Hadouken. The Mavericks come in ready to rumble, and X's response is basically "You know what? Screw you." Fireball Death'd!
Zero does it again in X3; this time he temporarily becomes a playable character.
Later on, it gets reversed. Vile and X tangle again. X loses again. Zero saves him again, but gets killed by Vile. This fills X with a righteous fury and makes him uber-pissed off. Bye, Vile.
The conversations between X and Sigma on the last stage of the first game. Made better by the voice acting in Maverick Hunter X. X's Unstoppable Rage at the death of Zero and his subsequent words make it that much more personal when he beats Sigma.
Sigma: I will show you the true potential of Reploids! X: I'm going to finish this! Right here! Right now! I WILL DEFEAT YOU, SIGMA!
What's even better is that this makes X's character arc, started from the Day of Sigma OVA, complete. Everyone's been telling him that he can't hesitate to pull the trigger when there's a threat that needs to be dealt with, but X has been grappling with indecisiveness and his own innate desire for peace. It's not until he loses his friend that he realizes what his hesitancy has cost, and that he absolutely needs to stop Sigma. The fact that he wants to is just icing on the cake.
The cutscene after the Final Battle in Mega Man X5: X and Zero, the latter dying and half his body destroyed, were impaled by a decapitated Sigma's Mouth Beam. Zero proceeds to destroy Sigma with one lastZ-buster shot. What makes the moment awesome? The spoiler tags tell it all.
Axl has another one, believe it or not, in the ending of Mega Man X7. Sigma, barely standing after being defeated, punches Axl through a wall, incapacitating the latter. Before X or Zero could move, Red appears and attacks them. Sigma, seeing his chance, attempts to possess Red when the latter came closer to him. However, Red, who was actually Axl in disguise, shoots him at point-blank range while his back was turned, and the single shot was enough to blast Sigma through a wall himself, and send him plummeting to the ground.
X's Badass Boast at the end of Mega Man X6, in which he claims that he has better things to do than fight Sigma over and over, but if he returns X will defeat him for good. Of course, then he goes and starts whining in the next game and we get Axl, so yeah.
Throughout the Day of Sigma OVA, X is reluctant to use violence when fellow Reploids are at risk. But after being forced to watch Sigma attack Zero and launch missiles on their own city, he snaps, screams with righteous fury, and claws at Sigma's eyes with a ShiningFinger, permanently scarring him. And this is after he was impaled on Sigma's sword.
People like to make G-Gundam jokes because X is voiced by Dommon's English voice actor, but something that doesn't get noticed as much is that Sigma has the same English voice actor as Master Asia.
(Hard Mode) In the final level of X8, Vile, in a last-ditch attempt, attacks one of the characters after his Boss Battle, separating the two heroes, forcing the other character to traverse the rest of the final level alone. When fighting Sigma, the latter grabs the character in an inescapable vicegrip. Just when all hope is lost, the other hero arrives (the camera is even in first-person perspective of the Big Damn Hero as he closes in on Sigma and his partner!) and saves his partner, and the Boss Battle resumes.
In one of X6's endings, even if he is Zero openly claims that the world is in good (X's) hands, and that X will always be the better hero between the two of them. This is quite impressive, since Zero doesn't even think of himself as a hero. A humble badass.
In the intro to X6, they show Zero's crash into the falling Eurasia colony in X5. That interpretation of the event was one of the few things that the game has done right, and even better than how they showed it in the original. In fact, the original (X5) didn't even show the crash!
As bad as the game itself was, X6 (especially if you don't unlock Zero) was one long Crowing Moment Of Awesome for X. Honestly, continuing with this characterization would have made the remainder of the X series much more tolerable. And Axl probably would be much better favored if (assuming Zero was kept dead) X now needed to take Axl under his wing, like Zero did when X was a rookie. It'd bring the whole thing full-circle rather nicely, and would set up the X of the Zero series much better.
From the flashback of Sigma and Zero from X4, Sigma (The Hero at that time) is fighting the Maverick Zero. This is a CMOA for Sigma at first, as he gains the upper hand. But then Zero quickly turns the tables by breaking Sigma's arm off with his own. Before that, he also tries to fight Sigma, who holds a lightsaber, WITH A GODDAMN PIPE!!!