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."
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.
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.
Also 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.