Donald O'Connor running up and back-flipping off the walls at the end of "Make 'Em Laugh".
Even better because it was all improvised. When he fell over from exhaustion at the end? That was real.
Meta example for O'Connor: after literally collapsing from exhaustion filming it, he found out that the footage hadn't turned out right. So, he did the whole thing over again.
Some of the other song and dance numbers, like "Moses Supposes" or "Good Morning", considering how intricate they are and how few cuts are used. Few laypeople realize just how much effort and practice has to go into making synchronized tap moves look so utterly effortless on screen.
Moses Supposes is two takes.
What happens at the end, when Lina's voice is revealed as a fraud.
The sheer glee on Don, Cosmo and RF's faces as they mime along with Lina's idiotic dance moves, right before they raise the curtain to reveal Kathy singing behind Lina.
Near the end of the "Broadway Melody" sequence, we see Don's character sulking after being rejected by his crush, again. Suddenly we hear a voice sing the exact same thing he did at the beginning. Then the two look at each other for a moment, the hoofer just shrugs and walks off. But Don's character has an epiphany and sings.
The sudden and unexpected appearance of Cyd Charisse's legs in that sequence and the dance number they lead to. There's a reason that shot is used as the page picture for She's Got Legs.
Gene Kelly improvised the iconic title number with a 100+ degree fever in one take. He also measured each puddle to make sure that the kicks felt right.
Debbie Reynolds was not a professional dancer, yet she managed to keep up with Gene Kelly and Donald O'Connor in "Good Morning" after only three months of daily practice. Long takes, too, so editing couldn't help her. She practiced until her feet bled and succeeded.