From the very first episode: Splinter and the Shell Cell.
Don't forget the part where the younger cop was shocked at Donatello giving them the stolen money, to which his older partner just rolls his eyes, as if it was a normal thing! Unusually Uninteresting Sight played straight there.
The Shredder ends up allying with the Turtles to stop Stockman during "Return to New York, Part III". But every time they seem to succeed, Stockman just comes back.
Leo: What do we have to do to stop this guy?!?
Shredder: (exasperatedly) I have asked myself that question many times...
In "The Shredder Strikes Back, Part One," during one of the scenes of the Turtles hanging out in April's apartment, we get this exchange while Raph and Donatello are watching TV:
Mikey: (Steps in front of the TV) May I have your attention please.
Everyone looks at him. He proceeds to cut loose with an extended belch. Raph and Donatello applaud him. At a nearby table, April looks disgusted. Splinter looks somewhere between resigned and amused.
Splinter: (To April) This is what I live with all the time.
Don: Say what you will about the Foot, they do not skimp on the headgear.
Also from "The Shredder Strikes Back, Part Two," we get the following one-liner from Mikey when the gang finds an avenue of escape:
Mikey: Ladies and gentlemen, turtles and rats, run, do not walk, to the nearest exit!
For all the drama and intense fighting of "The Shredder Strikes Back, Part Two," it does have quite a few funny moments. When the Turtles retreat to a walk-in refrigerator, things go quiet. Mikey wonders if the Foot ninjas retreated, leading to the following exchange:
Mikey: Kinda quiet out there...you think they all just went home?
"Careful with your weapons discharge! We cannot afford to damage the techno-organic equipment in this chamber.
"What did I just say?"
In "Notes From The Underground part 2" while the group is following Quarry off a cliff into a whirlpool:
Don: Following monsters off a cliff may not be the smartest thing we've ever done.
Mikey No, but it's definitely the grossest, that thing looks like a gigantic toilet!
During one of the episodes in the first season after rescuing Angel with Casey and returning her to her grandmother who offers him a slice of pie in gratitude, Mikey wants some. But his brothers drag him off telling him they are ninjas and they live by the shadows never to be seen.
Leo fights a bunch of Triceratons with a pair of plungers. After he beat them down, he kissed the plungers. Then, he expressed a disgusted look on his face after he realized what just did.
The Turtles asked the Fugitoid to build his teleportation device and Donatello make this very unnecessary comment:
Donatello: What's to stop them from ripping apart your robot body and download all the data from your brain?
Prime Leader: Slag! Now why didn't I think of that?!
Michaelangelo: Good one Donnie!
These moments that make fun of the original show;
"Secret Origins Part 3" has a particularly funny bit where April wears a yellow jumpsuit to try and disguise herself as a news reporter. When questioned about it at the end, she responds with, "Yeah, in another lifetime maybe."
"I hate walking on my tentacles!"
"Shut up, Krang!"
Mikey spotting the teenage mutant ninja rats (and their terrapin-like sensei) in the sewers during "Turtles in Space, Part 1 - The Fugitoid".
Also, Raph taunts the Shredder, who latches onto his head in response;
Mikey: Get it off! Or there's gonna be a tiny little Shredder alien busting out of Raph's chest! (Gets Dope Slapped) Ow!
An understated one comes from the 'Rogue in the House' arc, after Don encounters Stockman— or, rather what's left of him. Stockman tries to charge at him from within the jar... and Don jumps backwards, like he had some reason to worry. Aside from the The Smart Guys proving that they're Not So Above It All, the squeak that Stockman makes as he collides with the jar may be what makes the scene.
When someone attempts to assassinate Leonardo during the Battle Nexus tournament, he uses a makeshift weapon. The makeshift weapon in question? What appears to be The Shredder's helmet. The implications of it and Leo throwing it away in disgust are amusing.
"The Battle Nexus, Part Three": Raph's total freak out when Mikey beats him.
A small moment, during the episode "H.A.T.E." when Donatello and Raphael search the titular group's base for bombs the high-pitched fear squeak Donny makes when he sees the bomb in question is both hilarious and ADORABLE
Speaking of "H.A.T.E.", Casey's mom has the best ending line:
Casey's Mom: One more thing: you don't have to tell me now, but someday I'd really like to hear the story behind that big rat and the giant turtle.
And the looks Splinter and Leo give each other on overhearing this.
The Triceraton invasion gives us a few, not least:
"Hey! This is the Prime Leader's ship. Didn't we steal this once before?"
"And guess what? We're stealing it again."
Followed immediately by...
"Prime Leader! The, uh...the...the terrapin creatures have stolen your private space cruiser." (beat) "Again."
"WHAT? I just had it detailed!"
Earlier on, when Casey and April are organizing the human prisoners:
Casey: Alright people, listen up! No acting like people in an invasion movie, 'cuz they always get vaporized! Somebody always gets vaporized. Yo pal! It's probably going to be you! It's like you got a sign on your back that says 'Vaporize Me'!" (the other people back away from the guy he's pointing to)
Mikey has been shirking his training (and bragging about being the Battle Nexus Champion)... right when Kluh challenges him to a rematch. He freaks out and tries to get the others to help him train, resulting in:
Mikey: "Master Splinter! You'll help me, right? You'll train me?"
Splinter: "Training, Michelangelo? Surely there are comic books I could be reading, or perhaps I could go watch television instead. Ah, wait my son, I know! Perhaps instead of training, I will go eat pizza! Yes, pizza sounds very appetizing right now. With pepperoni! And karma!"
Mikey brings up being the Battle Nexus Champion for the umpteenth time, prompting one of the other turtles to comment that he's surprised that Mikey doesn't wear the trophy around his neck. He replies that he tried to, but the chain kept breaking.
"Beginning of the End" starts with Michelangelo standing on the bow of the Ninja Tribunal's ship;
Mikey: [[Film/Titanic I'm king of the world!]] (Lightning flashes, scaring Mikey and causing him to jump into Tora's arms) Mikey: Heh heh, at least there are no icebergs out here, right, Tora-san? Tora: I hope I never learn to understand you, Mikey-san. (Drops him)
Back To The Sewer
The turtles unknowingly befriending Hun online while playing an MMORPG.
The Reveal that the Ultimate Gamer was just a little girl.
The way Casey Jones screws up his attempt at proposing to April O'Neil.
Casey: I love you as much as Gretzky loves hockey, April, and I want to play you for the rest of my life. I mean, the rest of your life! I mean, Gretzky's life!
The Incredible Shrinking Sterling has a priceless moment when Sterling is lost in time and lands in the sewers and thinks he's back in the present only to see the toddler turtles mistake him for a toy.
Sterling: Oh, yuck! The sewers...Oh well, at least I've arrived back to present day.
The slight change that happened in the second version of the theme song:
Where we once got the menacing shot of Shredder on top of a building saying "Watch out for Shredder!" is replaced with Leonardo, Raphael, Donatello, and Michelangelo who are now in that same spot. It then shows Mikey holding Shredder's helmet as he danced around and shouted "We shredded Shredder!"