Stannis deducing that Arnolf Karstark is The Mole, getting him and his nephew arrested and bullying several Karhold men into fighting for him instead of House Bolton.
Stannis' speech after being warned of the danger of Ramsay Bolton.
Stannis: "I defeated your uncle Victarion and his Iron Fleet off Fair Isle, the first time your father crowned himself. I held Storm's End against the power of the Reach for a year, and took Dragonstone from the Targaryens. I smashed Mance Rayder at the Wall, though he had twenty times my numbers. Tell me, turncloak, what battles has the Bastard of Bolton ever won that I should fear him?
Arya finally killing Raff the Sweetling so very long after having added him to her list, and just to make it sweeter she does so the same way Raff murdered Lommy in A Clash of Kings and even makes him unknowingly echo Lommy's last plea for help.
A far more subtle one in the same chapter is that it reveals at the end that Bran's greenseeing has progressed to the point where he can now see Arya all the way from Westeros.
Jaime killing Aerys. At 17 doing the most unbelievable crime of all time. Killing a king, a king he was sworn to defend. And his reasons make him a total woobie (at least initially).
Stannis holding Storm's End for a year under siege, coming to within weeks of starving to death whilst the Tyrells and the Redwyne's feasted and wafted their foods' scents over the walls. Stannis did this while he was about seventeen, impressing Ned so much that he believed Stannis to be fearless.
Jon Arryn, knowing that King Aerys has summoned Ned and Robert to come to court in order to have them both murdered as well, refuses to turn over his young wards and instead raises his banners in revolt. Everyone should remember that it wasn't Ned or Robert who instigated the rebellion, it was Jon Arryn who risked everything to protect the two young men he helped raise.
Robert Baratheon and Rhaegar Targaryen meeting in single combat at the Trident. In an epic battle, Robert eventually managed to Drop the Hammer on Rhaegar and basically win the Rebellion right there.
Faced with the Queen's order condemning Nettles to death, the Lord of Maidenpool baulks at violating Sacred Hospitality. His loyal maester solves the problem by tipping off Prince Daemon, then telling his lord to arrest him for treason. Instead of handing him over for execution, the lord declares they're both traitors and changes sides.
The Caltrops killing the Two Betrayers - with Bold Jon Roxton walking alone up to Hugh Hammer and all his guards delivering a badass Pre-Mortem One-Liner before cutting him damn near in half (even managing to kill three of Hugh's guards as well before he was overwhelmed by the rest. Ulf is killed by a Taking You with Me from Hobart Hightower involving poisoned wine.
"History has little good to say about Ser Hobert Hightower, but no man can question the manner of his death. Rather than betray his fellow Caltrops, he let the squire fill his cup, drank deep, and asked for more."
Daenerys Frakkin' Targaryen. Dany is like a Crowning Person of Awesome all on her own after about the midpoint of A Game of Thrones, but here are a few highlights:
First, when she kills Mirri Maz Duur and hatches her dragon eggs, and WALKS INTO THE FIRE TO PULL THEM OUT and comes out unscathed (except for her hair)at the end of A Game of Thrones
Then, when she burns down the House of the Undying AND outwits the Qaartheen to escape after leading her people through a desert death march mostly intact during A Clash Of Kings
Finally, she turns her small group of Dothraki into a large group of warriors by freeing the Unsullied she just bought by ordering them to kill the men who owned them five minutes ago, then leading them on a rampage through the East, freeing slaves and leaving burning ruins of what had once been decadent, immoral, evil cities in her wake during A Storm of Swords. Put simply, she is the most badass fourteen-year-old in, well, EVER.
And this is after she shredded half his face off with the whip he just gave her.
Of course, this being a World Half Empty, after Daenerys accomplishes the events described above, it falls apart. But instead of pulling a Karma Houdini—or letting the author throw one on her behalf—she decides to stay and learn from her mistakes. For a girl of sixteen, that's another Moment of Awesome right there.
Drogon in A Dance with Dragons. If you've gotten to the scene, no further explanation will be necessary.
And her killing Shagwell the fool - "Laugh!" but Shagwell could only moan and cry.
"Sapphires." Finally, Rorge gets what is coming to him. Also qualifies as a CMOA for Gendry, who rids the world of Biter a few minutes later
No love for Sansa? In book one after realising what Joffrey really is she almost pulls a Taking You with Me and is only stopped by the Hound. Book two she saves Ser Dontos life which allows for her eventual escape from Kings Landing. And in book four she's showing real signs of Magnificent Bastard potential.
Her defiant refusal to kneel at her forced wedding to Tyrion was a brave bit of resistance in a scenario where she had virtually no capacity to refuse her captors anything.
The backstory featured a confrontation between Aegon the Conqueror's sister-wife Rhaenys and the Ruling Princess of Dorne, Mariya "The Yellow Toad" Martell. The Dornish had used guerilla tactics to destroy the invading Targaryen armies little piece by little piece, at the same time negating them a target for their dragons. When Rhaenys got fed up and flew straight to the Dornish capital the following (paraphrased) exchange took place:
Mariya: This is Dorne. You are not wanted here, return at your peril.
Made even more awesome by the fact that Mariya was an 80 year old woman, nearly blind, and she still dared to spurn a freaking dragon rider who had burned an army to cinders a few days back. The Dornish (even women)must have balls of Valyrian steel!
From "The Hedge Knight", many moments relating to Dunk's Trial of Seven:
The smallfolk all being on Dunk's side even though his opponents include three princes of the blood and three members of the Kingsguard because he was "a knight who remembered his vows".
Baelor Breakspear himself, Hand of the King and heir to the Iron Throne, volunteering to be Dunk's final champion, even though it means going against his own brother and two of his nephews.
Ser Humphrey Hardyng fighting at Dunk's side even though he already had a broken leg from his earlier clash with Aerion.
Raymun Fossoway being knighted on the spot to take Dunk's cause after his treacherous cousin Steffon betrays Dunk to fight for the other side. During the battle he gave his cousin a few bruises to remember and afterwards went on to found the cadet branch of House Fossoway based on his repainted "green apple" banner which would grow to become the equals of the main "red apple" branch.
Finally, his acceptance of Prince Maekar's offer to let him take his son Aegon as his squire- on the condition that the two of them roam the country living the low life a simple hedge knight and his common squire. When Maekar incredulously reminds Dunk that Egg is a prince of the blood, Dunk points out to the prince and now heir to the throne that neither of his other two sons ever learned humility or decency, so maybe it's time to try things a different way. Maekar doesn't answer, but Dunk gets his way. (And if you know about the history of the series, you know Aegon V grows up to be a fair, honourable and beloved ruler, so Dunk knew what he was talking about.)
There are several great moments for Dunk in "The Mystery Knight," but chief among them?