Yuki Nagato vs. Ryoko Asakura. Particularly the moment when Asakura has just shoved a bloody great glowy knifey thing through Nagato, and Nagato's only response? "Commence termination of data link." Check and mate.
Let's also point out that no more than a few seconds before, Nagato took a shot of five spears of steel right into the chest while protecting Kyon. Her ENTIRE RESPONSE is to stare at Kyon with the best confused look she can pull off, and speak in that completely deadpan voice, "You need to stay right where you are. I'm fine." She then calmly reaches up, and PULLS ONE OF THE SPEARS OUT. Like it's something she does every day. It turns into a desk.
More entertainingly, right after the fight, she slumps to the ground. Kyon, panicked, begins to talk about getting an ambulance. Yuki comments "No. My physical injuries are not critical." So, you know. This means taking five spears of steel directly through very, very vital parts in the chest, then two immense glowing spear... somethings. Which basically impale her right above the breasts. Read: lungs are FUCKED. These aren't critical injuries. Holy shit.
Some lines from Psycho-Asakura are simply terrifying and awesome:
"You don't wanna die? You don't want me to kill you? :("
"[...] because whether you care about it or not, I really want you to die! ^_^"
"Does it hurt? I'm sure it does. Better enjoy it while it lasts, because that'll probably be the last sensation you'll ever feel. ^_^"
The climax of the Melancholy saga, a.k.a. Kyon saves the dimensions by making out with Haruhi, essentially giving her The Big Damn Kiss. Not only was the tension built up perfectly, but the animation and music made the scene even more awesome. A Heartwarming Moment.
The Sigh of Haruhi Suzumiya
Yuki's actions during the "Mikuru Beam" incident in Sigh are also pretty epic: she catches laser beams. Barehanded. Realistic laser beams at that. Meaning that, yes, she is actually capable of moving faster than the speed of light.
The Boredom of Haruhi Suzumiya
The Cave Cricket battle, which showed off Koizumi's skills in battle as well as more Nagato fight scenes. Too bad it was short, but the BGM to the fight made it all worthwhile.
Kyon gets one by utterly demolishing Koizumi's story, demonstrating beyond any doubt that just because he's the only normal person in the group does not mean he can be treated like a child. Bonus points for his very first argument being "Haruhi wouldn't really wish for anyone to die."
The Disappearance of Haruhi Suzumiya
Mikuru punching Kyon in the face counts as her CMoA. Kyon was clearly in a desperate situation, but bringing up the mole on her breast was still a terrible idea. And then Mikuru showed us that even the resident Moe-blobhas her pride!
When Kyon finally finds Haruhi outside the Koyouken High and uses his trump card on her. Haruhi: - (noticing) North High... What's your name? - Kyon: - John Smith. Haruhi faints. She just couldn't stand the awesome. Before that, we were pretty sure nothing could surprise her that much.
At the English-language premiers of the film that line has elicited standing ovations from the audience.
Kyon's monologue about wanting to be with the SOS Brigade.
Ryoko stabbing Kyon extremely awesome. The movie has done a nice job portraying sheer abject terror that spreads across Kyon's face as the camera focuses on Ryoko Asakura, back from hell with a casual smile.
Kyon told Yuki to tell the Data Overmind that she was under his protection, and if anything happened to her, he would destroy it, alien god or no... with four words: "I am John Smith." Also a Heartwarming Moment.
The official translation gives Kyon a line that makes this even better. ("Him" refers to the Data Overmind, obviously.)
Kyon: Tell him to suck it.
The subtler bit just after, where he tells it to allow Yuki to act more human, and she thanks him, is also a good candidate for the moment of the book.
Kyon's future self also has a Big Damn Heroes moment when he leads Yuki Itsuki and Mikuru to rescue his past self. Haruhi may unintentionally summon the weirdness... Kyon can mobilize it.
The Rampage of Haruhi Suzumiya
When Endless Eight... well, ended. In the anime, anyway. Although the loop got pretty torturous, if they hadn't done it that many times then Kyon's solution would have been more of a Mundane Made Awesome moment, but I nearly cheered out loud. That and the epilogue also border on Heartwarming Moment... somehow.
It borders on that for one very important reason...Yuki. As the episodes go further along, she becomes more and more visibly bored, which for her, is the equivalent of a hair-pulling fit of despair. When Kyon does his Epic Loop-Breaking Speech, her eyes actually LIGHT UP. She still doesn't say anything, but the look on her face makes it so adorably worth it.
Day of Sagittarius
Haruhi hearing the Computer Society President ranting about a "DUEL!" from a distance and delivering her famous Dynamic Entry. That was pretty damn awesome as well as hilarious.
In The Day of Sagittarius Kyon talking Haruhi out of betting female brigade members on a computer game duel counts. He suggests she bet herself for a change
Yuki gets another one, at least in the eyes of all the Playful Hackers out there, when she hacks Day of Sagittarius III's game code in the middle of the game to turn off the Computer Society's cheats. Moreover, on Kyon's instructions, she does it without any superpowers, just her native intelligence and Rapid-Fire Typing.
Itsuki Koizumi gets such a scene at the end of the novel story Snow Mountain Syndrome, when he declares to Kyon that should his Organization ever come into conflict with them, he will betray his superiors to side with the SOS Brigade. Really now, several of the scenes in the later novel stories reinforce the concept of the SOS Brigade as True Companions.
Yuki answering to her past self why she didn't synchronize: "Because I don't want to."
The Surprise of Haruhi Suzumiya
Ryoko SAVING Kyon from an interface even more psychotic than her. And in true Ryoko Asakura fashion, she does this in part by holding a knife to Kyon's throat and threatening to off him herself! She doesn't do it, thankfully, but still a cool moment.
At the climax, Kyon does a Diving Save out the clubroom window to save Haruhi from falling to death.
In the same book, around the same time, Itsuki pulls a Let's Get Dangerous! moment on Fujiwara and fights him. After Kyon's abovementioned moment, he witnesses an explosion from the room the two were fighting in. All that's said of it afterward in the book is that Fujiwara "didn't end well."
And how does Kyon's moment above get resolved? He and Haruhi are caught by a Celestial. Which was summoned by Yasumi, Haruhi's alternate self. That's right, Haruhi unconsciously pulled a Big Damn Heroes moment for herself.
Kyon blackmails Haruhi, with the threat of calling her "Haru-Haru" if she forces Mikuru to dress in humiliatingly revealing clothes at a concert... Haruhi's reaction was priceless and must be heard to be believed. Just goes to show, whether you are an omniscient alien or an obliviously omnipotent schoolgirl, Kyon knows how to blackmail. (At 2:47 of this clip)