Solidus taking on THREE RAYS at the same time, and destroying them all within a matter of seconds. And he does this with a P90.
Solid Snake's stupendous character traits are comprehensively demonstrated when you rush in with the computer virus, covering a range from "We're counting on you" to "Change sides? I don't recall ever saying I was on yours."
Supposedly this is why Hideo Kojima had Raiden be the playable character for most of the game... because it was only through a rookie's eyes that you could really feel the awesome.
In MGS2, Ocelot steals Fortune's electromagnetic device which was the cause of all projectiles missing her, not supernatural powers of luck as she believed, after shooting her in the heart. But it is revealed her heart is located on the right side of her chest, she stands up, and as Ocelot launches a massive amount of missiles at her, she raises her hands, and despite the device being missing, the rockets all misfire anyway. Then she dies, knowing she will see her family again.
Ocelot's backstabbing also counts. From that point on, he became undeniably one of the greatest Magnificent Bastards in all of fiction.
Regardless of how other fans perceive Raiden, to some, he established his Badass cred in MGS2. That no one has bothered to mention how he took on multiple RAY units, three at a time, before the final showdown with Solidus in a sword fight? Heck, an option he has available to make his fight with the RAY units even more Badass is taking them out with the Grenade Launcher instead of the Stingers! For being a rookie compared to Solid Snake, he can hold his own!
Sometimes it's the simple things, like the Power Walk moment in MGS2 with Otacon (and his sister's parrot) leading Snake and Raiden. Or Snake and Otacon's bro-tactular fistbumping right before it.
It's made even more awesome because of how competent Snake's AI is. While he does require support, it never feels like you're babysitting him and he'll likely equal you in kills. For an early PS2 game, that was impressive.
Emma's computer virus. She designed the virus to corrupt an AI that she can barely confirm the existence of, and has undoubtedly never once had access to. On top of that, she manages to manually upload it while dying of a fatal stab wound.
In the Tanker mission, Otacon will eventually mess up one too many Chinese proverbs and Mei Ling cuts in to correct him.
A quieter one for Snake. Usually, during Emma's Escort Mission, calling Snake on the codec would result in hearing a fight with Emma and Otacon. However, the final call has Snake talk to her instead, due to Otacon being busy. After listening to her rant for a minute, he then promptly begins to quietly call her out, and effectively tells her to quit her whining.
Snake: Of all the idiotic—
Raiden: I think that's enough.
Emma: Idiotic?! It's my goal! My reason for being alive! Hal is wrong! He is only being used by his country! He's pitiful! Even you guys are using him! Me... I'm using science to achieve my own dream!
Snake: Real bright, lady! That's exactly the line that scientists are never meant to cross!
Emma: Right! I sold my soul for revenge! I'll never forgive him!
Snake: I'll pretend this conversation never happened.
Emma: No! You tell my brother exactly what I said! You...
Snake: You're speaking your mind... not your heart.
Otacon and Snake piloting a helicopter with Raiden providing ground firing support versus a Harrier Jet is easily the most awesome fight ever.
There is also the slaughtering of two rooms of ninja-like Arsenal Tengu soldiers back to back with Solid Snake inside Metal Gear with a katana.
Snake breaking his handcuffs and diving in after Liquid!Ocelot and Metal Gear RAY.