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The special effects of the first movie (Done by Jim Henson, the man behind Sesame Street and The Muppet Show, of all people) arguably make the entire thing one long Moment of Awesome, but what stands out the most are the fights between the Turtles and the Foot which are energetic and imaginative. The fact that they pulled this off with four of the principal performers in 100lb costumes and nearly blind makes it all the more impressive.
Casey Jones telling off the junior members of the Foot after claiming that they're a "family."
Before that, he takes a beating from Tatsu which Casey then turns into a Curb-Stomp Battle once he gets his hands on a golf club.
Splinter having the Shredder at his complete mercy with a simple flick of the wrist.
And Casey finishing him off in the garbage truck complete with a deadpan "Oops!" when he turns on the crusher.
Michelangelo's nunchaku duel with the Foot Clan member.
The Turtles first real fight with the Foot Clan doesn't go their way. Raphael is injured and out of the fight, they're taken by surprise, and they are slowly being overwhelmed. It's only through the arrival of Casey Jones and a quick retreat that the Turtles survive. The rematch, however, is far different. The Turtles manage to not only hold them off, but force the entire Foot Clan to retreat. And keep in mind that their first bout at April's was only against a small contingent, while the rematch was against the entire clan!
The best part of the the battle in April's apartment is when Casey shows up to cover the Turtle's escape and wipes the floor with the entire squad.
And then The Shredder himself decides to join in, and utterly thrashes the Turtles throughout the fight with only Leo managing to get in a rather minor hit. It takes Master Splinter himself to finally bring the man down through misdirection and trickery. Say what you will about his cartoon incarnation at the time, THIS Shredder was nothing short of a badass of the highest caliber.
Leo gets another one in TMNT after being knocked into a glass case full of giant ancient blades by one of the stone warriors. The scene cuts to him standing up, with no less than five of the swords on the hilts on his back while he's wielding two of them. "Come to daddy."
And before that, the rooftop fight between Raph and Leo. After over twenty years of existence, we finally get to see a no holds barred fight between the two brothers, one that at first seems to be a draw before Raph manages to break Leo's blades and come a hair's breadth from killing him. While the fight itself is arguably the emotional climax of the film, the look on Raph's face and Nolan North's performance manage to make it a truly devastating, yet still awesome, fight.
Another one; Leo was owning Raph before The Reveal of Raph being the Nightwatcher, completely devestating him in the fistfight, it was only after Raphael's helmet was knocked off did Raph manage to even up with Leonardo, which means... Leo was holding back. And, in an actual fight to the death (as shown in the ScrewAttack video), Leo would've won.
It may have been that, or it may have been Raph's switch from his Rule of Cool chains to his sai, the weapons he's trained with his entire life, that let him even it out.
The animation crew at Imagi studios really deserve major credit for that sequence, the expressions as noted were incredible; especially when Raph realizes that he may have actually done the deed coupled with Leo's own harsh look of "See? this is why you're not ready," as Raph's sai is in his face.
While The Second is Lighter and Softer, The Shredder gets a moment when at The Foot Headquarters Tatsu challenges anyone to face him for leadership only for Shredder to walk through the door and states that he challenges and Tatsu and The Foot's reactions to his return.