In episode 20, he manages to defeat the Monster of the Week who was stealing people's love in a awesome way. The way he walks into the battle only adds into awesomeness.
In episode 1, chief Odagiri's Establishing Character Moments: first, when the Vyram attack the orbiting Earthship and Ryu won't evacuate due having just seen Rie being sucked in the void, Odagiri tries to reason with him and, when he won't move, she punches him out (remember, Ryu has already been turned in a Jetman and is superhuman, she isn't), drags him to the Jet Hawk, and pilots it to safety with both on them on board; hours later, when Ryu finally wakes up, she slaps him out of his desperation, conforts him, and gets him to accept he can fight the Vyram and avenge his girlfriend.
Ako showing just how superhuman the Jetmen are: she promises to break the high jump world record, and when taking part in a contest she does just that, jumping 2.40 meters without even trying. For comparison, the female record (standing since 1987) is 2.09, and the male one, standing since 1993, is 2.45 (not beaten by Ako, but then she wasn't really trying). And she's still a high-school student.
Ryu taking on a giant jet monster on foot, by himself, and winning. True, he did wound it with his jet earlier on, but its still damn impressive.
The dogfight between Ryu and the jet monster. When the monster snatches up Gai to have a hostage? Ryu quickly targets its arm and flies by it as it's firing everything it has at him, slicing into it's arm to make it drop Gai and send it crashing to the ground.
How Radiguet makes the monster: he sees Ryu flying around in the Jet Hawk and Odagiri flying as his wingman with a normal jet, and, out of the blue, puts a Dimensional Bug in Odagiri's jet to turn it in the monster Fighter Dimension.
Faucet Dimension's first target is the chief, who gets attacked while under the shower. She escapes without much trouble.
The immense can of whoopass the Jetmen open on Faucet Dimension in their first battle as an actual team.
Episodes 4 and 5 are basically Kaori becoming awesome.
Episode 6: Ryu's grandmother forces Mirror Dimension to release his captured victims by jumping in front of his capture beam and using a mirror to deflect it into Maria, trapping her and forcing it to release everyone to let her out. The Jetman team are impressed.
In episode 9, Raita frees his childhood sweetheart from Fashion Dimension's spell, shocking Trans. When the team starts beating the monster down, Raita comes up late. Why? He was busy getting his usual giant rocks...four of them which he uses to pound the living heck out of the monster.
Radiguet: Tranza, Try calling out my name. Tranza: (Shouting in pain) R-Radiguet! Radiguet: What did you say?! (Stomps on him hard) Tranza, Try calling out my name! Tranza: R-Radiguet! Radiguet: What?! Just Radiguet?! (Intensifies the pain towards Tranza) Tranza: (Shouting in pain) Lord Radiguet!!! Radiguet: That's Right! (Pulls the sword out of Tranza's palm) I won't kill you. You will fear my name your whole life, while living among the humans!
Episode 43: After Vyram nearly killed her and the Jetman, Commander Aya pulls a Big Damn Heroes to save Ryu, Gai, Raita, and Ako while they were in the Jet Icarus while she is in the Jet Garuda. Aya defeats the Bio Drill Dimension all on her own in a Curbstomp Battle. Now this is why Aya should have been an official Sixth Ranger!
She was also the one who managed to get Ryu out of the Monster of the Week's spell that made him lazy in the aforementioned episode 11.
There's Episode 3 when she's giving most of the team Training from Hell and has them attack her, curbstomping all three. Yes, they're untrained, but they've also got superstrength while she's just a Badass Normal. Later, when Facet Dimension attacks her, she manages to save herself. By cutting it's tendril off.With a razor.