The few seconds, at the end of The Bourne Identity when Bourne takes the mook's gun, and ends up shooting it upside-down with his pinky, Guns Akimbo to nail the second one, then throws the dead body over the rail, jumps on top of it, and headcaps the third goon while fucking freefalling, and hits the floor eight stories below, with the body as a pad, and then gets up. Admittedly, he's still very badly injured by the fall, but it's awesome nonetheless.
He nearly killed an knife-wielding assassin, in an apartment, WITH A BOX OF SCR- with a pen.
In Identity, Bourne reads a roadmap of Paris for about a minute, and in that time, manages to memorize every road, alley and sidestreet like he lived there all his life. He notes the location of a car park, then outdrives the police in a beat up mini-cooper.
He had been stationed in Paris for a while as John Kane, so the map probably jogged his memory of the layout. This does not in any way detract from the awesomeness of this scene, of course.
All of Jason's I Can See You lines, "Get some rest, Pam, you look tired."
Another call ender by Bourne: "That's easy. She's standing right next to you."
The final car chase, one of the most brutal of the decade. Bourne using a crapped out Russian Zil taxi to successfully outwit the Moscow police, the FSB and use some very nifty Car Fu to kill Krill who is driving a larger, more powerful Mercedes G-Wagen. What's more, it all takes place in rush hour traffic while Bourne is attempting to stabilize a gunshot wound.
The CIA agent Jason knocks out while he's being held in Naples? Matt Damon actually knocked out that actor for real.
In The Bourne Ultimatum, when Bourne kills a Treadstone agent with a bathroom. Yes, the entire bathroom. Although, honestly, everything Bourne does in the entire series probably qualifies for this trope. What a badass.
The scene near the start where Bourne is trying to maneuver the Journalist to safety by talking to him on his mobile, while periodically beating up mooks. It wasn't Bourne's fault that the journalist was Too Dumb to Live.
From Ultimatum, said by Pamela Landy to Noah Vosen: "You'd better get yourself a good lawyer." Enough said.
Another phone/distraction awesomeness: When Jason and Nicky prepare to leave Daniels's safe house with Treadstone agents closing fast, Jason phones in to police that there's gunfire in the area, fires several rounds in the room, and takes Nicky downstairs in a quick, casual manner. They walk to Nicky's car without breaking stride while the arriving Treadstone agents bump into a police squad responding to Jason's call, allowing the duo to easily disappear.
This troper prefers the ending, Where you see a single gunshot and Bourne fall off of the roof into the river. You think he's dead, and Nicky is watching the news report. It gets to the part "police have been searching for 3 days, but so far no body has been recovered". She smirks because she knows he's alive.
And the traditional Bourne ending song, Moby's Extreme Ways starts up right then, like a Scare Chord/Crowning Music of Awesome hybrid. That was definitely the best startup for that song anywhere in the series.
Aaron Cross taking down a UAV with a Sniper Rifle in Legacy, while still getting over the fact that he'd been betrayed.
Marta kicking the government Terminator to his death.
Cross timing Marta firing a gun so he could identify the movements of the last agent and shoot him through the wall.
Moby's "Extreme Ways," played over the end credits of the first two. The remix for the third movie is even better.
The main theme slowly evolves over the series too. The third film undeniably perfects it, with Jaws-like simplicity too.
It was remixed again for the fourth movie. Just as awesome.
"Treadstone Assassins" is quite nifty also.
Special mention has to be given to the track "Bim Bam Smash" on the Bourne Supremacy soundtrack, particularly the last 2 minutes of the track. It accompanies the entire chase at the end of the film, as well as, the car fu battle between Jason and Krill. It's not very often that music accompanies a scene so perfectly but this score manages to do it and convey the tension of the scene.