A very quiet Moment Of Awesome happens when everyone takes Vimes back to his room after he gets drunk and Angua notes how spartan the place is. She finds a book with a list of female names in it along with monetary figures. Her conclusion is that he spends all his money on loose women. Carrot very quietly, deliberately, and coldly corrects her. He's been giving all his money to the widows and children of deceased coppers. Doubles as Moment of Awesome for Carrot too.
"If you have to look along the shaft of an arrow from the wrong end, if a man has you at his mercy, then hope like hell that man is an evil man. Because the evil like power, power over people, and they want to see you in fear. They want you to know you are going to die. So they'll talk. They'll gloat. They'll watch you squirm. They'll put off the murder like another man will put off a good cigar. So hope like hell your captor is an evil man. A good man will kill you with hardly a word."
Carrot's "If you decline I will have no choice but to follow that order" line to Dr Whiteface. As it says, "Sergeant Colon was lost in admiration. He'd seen people bluff on a bad hand, but he'd never seen anyone bluff with no cards."
For those who haven't read it, the order is if Dr. Whiteface refuses to answer questions or asks him to leave, he has to leave.
'Listen! If I shout,' said Dr Whiteface, going red under his makeup, 'I can have a dozen men in here.' 'Believe me,' said Carrot, 'that will only make it easier for me to obey.'
The "fridge scene", where Detritus went from the dumbest Troll in Ankh-Morpork to a friggin' supergenius
Cuddy gets one for getting Detritus out of said fridge, and even going into Quarry Lane to hopefully find a troll doctor.
Corporal Nobby Nobbs gets one simply by virtue of how terrifying he makes the thought of him wielding a Klatchian Fire Engine (which is essentially an honest to God flamethrower!). I swear, for a guy who is walking comic relief, the sheer pyromaniacal glee he displayed over the possiblitly of using that thing was legitimately scary, and considering Nobby is a pathetic kleptomanaic Plucky Comic Relief most of the time, that's pretty impressive.
That's nothing compared to the sheer amount of knowledge Nobby has about any form of weaponry. He did serve as Quartermaster one time but he sold all the weapons, resulting in them losing the war (and Nobby winning as he joins the winning side).
Isn't it arguable that Nobby's real Moment Of Awesome is actually getting ahold of that armory in the first place? He uses his impressive shameless manipulation and crooked knowledge of regulations to mount one hell of a Bavarian Fire Drill.
Minor meta-Moment of Awesome goes to Sir Terry, simply for the awesome (and very creepy) description of the Klatchian Fire Engine.
Practically invoked by Vimes himself via inner monologue ("He'd like to take this moment and press it in a big book so he could look at it later") when he lets drop to Dr Cruces that the reason why he's acting like he owns the place is because he does.
Also arguably later as Vimes is wrestling with the Gonne and it goes off, hitting the ceiling, and he responds to assassins crying "who are you"—
And then Carrot picks it up, says "It's only a device", and smashes it against the wall.
One of the running gags is Detritus's trouble with saluting—each time he does it, he knocks himself out. At one point, he and Lance-Constable Cuddy are escalating into a fight, while Carrot's trying to calm a group of dwarves and a group of trolls. When he finds out his message of peace and tolerance is being undermined by Detritus bouncing Cuddy on the ground while Cuddy bites his leg, he realizes that he can't discipline them separately without re-inciting the riot. So what does he do? Order Detritus to salute, knocking him and Cuddy out at once.
Gaspode's moment with Big Fido, whose ideas and doctrine are “straight out of Mein Kampf” and who has the strays of the city completely under his thumb. Fido orders his wolf-pack-cum-killing-machine to attack Gaspode, at which point Gaspode, a tiny, scruffy terrier with every doggy disease possible, defeats them single-pawedly:
The full scene homage to For a Few Dollars More at the climax. Vimes' retirement watch even seems to be playing the same tune, if you read the bings out loud. Which is also one itself, that Pterry managed to accurately reproduce a song in text using nothing but italicized bings.
May also be a terrible pun, if you think about it. Heel, Hitler?
Detritus discovering his role as Drill Sergeant Nasty and managing to turn a crew of twenty odd Trolls, Dwarves and a few odd humans into a police force with some shouting.
When a massive riot breaks out over the Day Watch arresting the wrong troll, Carrot patiently waits until one of the Day Watchmen drops by and gives Carrot enough proof to show order has broken down, allowing him to form the Night Watch(who were recently dismissed) into a militia. He then proceeds to awesome his way up and down the street, recruiting trolls and dwarves into the watch simply by using their own animosity towards each other to both get them in and keep them in the watch and successfully restores order, while also figuring out how The Gonne was stolen.
Also during that when Captain Quirke of the Day Watch tries to stop them, Carrot calmly asks him what evidence he had of Coalface, the troll who's wrongful arrest started this whole mess, Quirke's defense boils down to "He was a Troll, I needed no bleeding reason to arrest him other than that" Carrot calmly responds by decking him and then getting back to work like nothing happened.
After spending a few moments worrying if he used "minimal necessary force". In Colon's words: "Is he still breathing? [...] Seems sufficently minimal to me, sir".
Carrot's meeting with Lord Vetinari, which doubles as a Crowning Moment of Funny, as they discuss new arrangements for the Watch, Carrot's lack of desire for the throne, and conspire to promote Vimes to Sir Samuel Vimes, Commander of the Watch.
They exchanged glances. The Patrician got the best of the bargain, since Carrot's face was bigger. Both of them were trying not to grin.
Really, the whole book is an escalating series of Moments of Awesome for Carrot.