The Beginning

An early one: the Initiates versus the Lesser Nevermore. The Initiates totally destroy those birds!

Bonding in Beacon

Ayers calmly walking up and headbutting Gaston. So awesome that he made Gaston respect him!

One for Ozpin. He catapulted the students into Initiation!
For Pius, there is taking out a half-dozen in Beowolves as soon as he hits the ground, at a combo speed that is comparable to Ruby and Yang in the RWBY canon.
Team DRKR being the first to kill their boss Grimm by knocking it out of the sky, blinding it, and stabbing it.
The second boss kill. Team JUMP literally melts the Kaijin's face off with the Power of Rock.
Kurira. Holy shit. Based off of Godzilla, he's the biggest Grimm ever faced by the characters, and so powerful that the only choice for the initiates is to Run or Die.

Jack and Remus have a brutal brawl, ending with Jack giving his respects to Remus as leader.
Followed by Gaston and Jack having an even more brutal brawl, which ends when the teacher must step in.

Whenever Alice fights, she stops attacks dead, throws her opponents like ragdolls and force chokes them.