Abby, completely unarmed, attacking an armed and fully armored KNIGHT who's ready to slay the Dracorex.
Abby: Oh, I'm a demon alright.
And when it becomes clear that the knight would kill her, Abby bares her neck, showing her clear willingness to die to protect the Dracorex.
Abby (again) driving a mobile home through backroads over the cliffs of Northern Ireland while a giant hungry Wyrmnote a 20-foot labyrinthodont is sitting on top of it trying to eat her.
Abby dodging a nuclear torpedo while driving a submarine backwards through an anomaly.
Also Abby giving a somewhat gratuitous, but still thoroughly entertaining catfight against Caroline in the Second Season finale.
Exhibit D:Cutter punching a Deinonychus. Okay, it didn't end the fight, but come on, he PUNCHED a RAPTOR.
In the first season finale, Cutter takes out a future predator by grabbing one of its babies, running into a greenhouse, and then proceeding to shoot out the windows to confuse the hell out of the predator, before shooting it. The calm, determined look on his face makes it so much more awesome.
Adding to the awesome is why he shot the glass. The future predator hunts with echolocation, and Cutter's trick created a sheet of falling glass shards that scattered the sonar every which way, blinding the predator and allowing Cutter to line up his kill shot. Yes, he improvised SONAR Chaff.
Cutter has also punched out a caveman. ...Okay, it wasn't a real caveman, but come on.
Cutter giving Helen a taste of her own medicine by toying with her emotions so she'll help him when handling Leek's guards. And then mocking her afterwards.
Cutter: Oh I'm sorry honey, I didn't mean it.
Exhibit E: Quinn using a helicopter to bait a Giganotosaurus into chasing him through an open wormhole. Best part? He Indy Ployedthe whole thing.
Claudia hitting a tiny bloodthirsty pterosaur with a golf club, despite being blinded and covered in blood. Considering that she's normally a noncombatant, it was pretty impressive.
I saw it this way: Human without weapon? Easy prey. Human with weapon? The deadliest thing in all creation.
The gorgonopsid (well, a gorgonopsid) gets its Crowning Moment during the first season finale. A future predator follows the team through an anomaly into the Permian. It wipes out the soldiers accompanying Cutter and Helen, and then a gorgonopsid shows up. Everyone is expecting The Worf Effect...and the gorgonopsid promptly subverts the trope. It takes every blow the future predator throws at it, and it just gets angrier, until it finally grabs the predator by the arm, crushes it (the predator, not the arm) and drags the carcass off to eat. WIN.
Becker hiding inside a car and then blasting a predator from below just as its about to pick off the rest of the team.
Jenny gets one in her reappearance in Series 4, when she defends her fiancee from a ravenous Hyaenadon, using nothing more than a spear, while in her wedding dress. And she has a nice little moment of badass rhetoric afterwards too...
Even Badass Bookworm Jess gets one in Series 4. In fact, since she's only nineteen and yet generally agreed to be one of the best possible team co-ordinators the ARC could hire (not to mention her complete non-chalance when faced with the creatures the ARC has to deal with), it could be argued that her entire presence in the series is a Crowning Moment Of Awesome.
In particular, her moment in Episode 6 is pretty amazing, where she shows herself to at least have some proficiency with firearms, and even saves Becker's life by disarming a bomb set by series Big Bad Ethan...
She gets another in 5x04, when she one-handed deadshoots a two-inch beetle from six feet away while half-conscious, feverish, and hallucinating due to a severe allergic reaction.
On a similar note to the above, Connor braining a very angry raptor with another big stick. Considering that he'd just woken up after being exposed to a stun grenade and falling out of a tree...
Matt fends off a Treecreeper (intelligent apelike arboreal theropod) with a tiny dagger.
Season 5, episode 5. Crowning moment for the series because finally, after five seasons of paying its paleontological dues by showing Seldom Seen Species, suddenly, right there in the Cold Open, a Tyrannosaurus rex in downtown London.
Matt's Crowning Moment: channeling the spirit of Stephen, he trips the T. rex with some Car Fu, and then tranquilizes it, a lot, from about one foot away.
Lester in Episode 5 of Series 5 when he sees a Treecreeper climbing on his car while everyone is called out.
Lester: (After seeing an Anomaly open in the ARC car park) Oh my God! My new Jag's in the car park! This time its personal.
Lester reappears a few minutes later walking out of a lift with an EMD and watching Treecreeper climb on his car.
Lester: OI! Have you ANY idea how much that thing cost?! (proceeds to repeatedly zap the dinosaur).
Also awesome, a few minutes before, he'd offered to help out by going into the field.
Abby and Connor survived in the Cretaceous. For a year. With minimal supplies and training.
In 2.04, Lucian (the victim of the week), hits a mer with a rather large rock via ricochet.
Lucian: Two points!
Abby stares down a sabre-toothed cat in 2.07.
Abby: It's trying to decide whether or not I'm a threat.
Connor: And if you're not?
Abby: (without a hint of fear in her voice or expression) Then it eats me.
Steven redeeming himself by taking on the Silurian Scorpion on his own. He doesn't kill it, but he impales it with a spear gun that pins it to the bottom of an underpass. He steps away and calmly takes out his phone, and the scorpion rights itself and prepares to attack again...but he's out of reach. He tethered it to the cement column with the spear cable.
A more tragic example: When he locks himself in with the creatures, stepping backward slowly into the middle of the room and giving Cutter a calm farewell, before looking the circling creatures levelly in the eye and closing his eyes.
Oliver Leek gets a Villainous Moment of Awesome when he sends all the ARC's security off on a wild goose chase, leaving Lester alone and unprotected. Leek then broadcasts his face on the Anomaly Detector's screens just so he can personally tell Lester to go fuck himself.And then sics a Future Predator on him.
His profiles and general onscreen composure lean him as a No Respect Guy, and thus he makes sure everyone takes him dead seriously as his plans go in motion (Cutter even outright admitting he didn't give the little weasel nearly enough credit). This is particularly apparent with Helen, who despite being the right hand girl of the scheme, spends the majority of their team up gleefully threatening and kicking him around, when she attempts to put him in his place as he makes a "dinner theatre" out of the team, she is very bluntly rebuffed.
Leek: You do not give me orders! No one does!
Cutter's confrontation with the Cleaner in the Silurian Desert in which Cutter tricks the Cleaner into drawing in a giant scorpion which then kills him.
Cleaner: You follow me, I'll shoot ya! Cutter: Well we're dead anyway. (Cleaner fires shots at Cutter's feet) Cleaner: I'm warning you! Cutter: You think I'm scared of you? Cleaner: You should be . . . (Raises gun. Cue scorpion.)
The pilot of Primeval: New World has Connor breaking into Evan's office, taking his Anomaly Detector and then bluffing his way out by telling Angelica Finch he's there to fix the internet connection, and all without any need of backup.