The whole 'Keep walking!' scene. Fievel keeps stopping the line entering the boat to point out other things, only to be told to "Keep walking!" Papa finishes the scene with, "This is the last time I take you to America!"
Seagull 1: Ach, I got a cousin in America.
Seagull 2: Ach, you and everyone else!
What makes this scene even funnier is that beforehand Papa is the only one telling Fievel to keep walking and everyone, including the band, is starting to grow steadily annoyed and when Fievel stops to point out the seagulls, they all yell at Fievel, "KEEP WALKING!"
3 mouse kids getting seasick and another offers them a bite of a pickle.
Tiger thinking he's playing rummy, but it's actually poker.
This bit by Tiger "You lost your family? Oh, dear. That's terrible. I lost my family, too. Years ago, I mean. [Starts sobbing] Eight brothers. Ten sisters! Three fathers!"
When Tiger reveals why the other cats think he's weird.
Tiger: Listen, I like mice. (Fievel's crying gets worse) Oh, no. Not like that. You see, I don't eat red meat at all. I'm a vegetarian. Oh, a little fish now and then, but what I really like is a nice piece of... shh... broccoli.
When Tiger is being introduced as Fievel's guard, he brags about his abilities ending with, "...I have the instincts of a cat! Wait a minute, I am a cat."
Gussie Mouseheimer makes some funny bits too, mostly from the confusion her Elmuh Fudd Syndwome causes. At the 'Wawee', she asks what they're fighting for, and is told "FWEEDOM!" by the assembled mice.
Gussie: Money is not evewything. I know, because I have money, and I have evewything.
Gussie: Awe we men, or awe we mice?
Fievel Goes West
When Tiger was lost in desert looking for Fievel.
Tiger: I'm lost and all alone, in a million acre catbox.
Before that, when he saw the mice running in fear, he swore to himself he'll be tough and brave.
This dialogue (After Tanya's had a tomato thrown at her due to her singing):
Tanya: Someday I will be a big star. People will come from miles around. Fievel: Yeah, to eat.
When Fievel and Tiger are lost in the desert, they both bump into mirages of Fievel's family and Miss Kitty respectively. They see each other in the distance and are overjoyed...until they decide that it's another mirage. When they pass each other...
Fievel: Hi, mirage of Tiger.
Tiger: Hi, mirage of Fievel.
During the Training Montage, Tiger is doing some tricks with a lasso, jumping through his own hoop. Then he does it again, and again, getting faster and faster until he ties himself up!
The dog chase scene. Poor Tiger can't catch a break when he's trying to get to the train station to say good-bye to Fievel, boards the train only to discover a dog's the conductor, then ends up on another train only to discover another dog, and when he falls into the water, he gets confronted by a dogfish. His deadpan response sells it:
Cat R. Waul luring the mice out west with a cowboy puppet is hilarious—John Cleese doing a cowboy impression is truly legendary.
The Treasure of Manhatten Island
Tiger decides to guard the subway entrance:
Tony: What are you, afraid?
Tiger: Who, me? No, I just figured it would be better to stand guard up here, and... protect your flank!
During a montage where Fievel, Tanya, and Tony show the Native American character Cholena around New York, on the Tunnel of Love at Coney Island, Tony puckers up to kiss Cholena, but she picks up Fievel and has Tony kiss him.
The Mystery of the Night Monster
Fievel draws a picture of Reed Daley with huge ears. When Reed complains, Nellie quips "For a more accurate likeness he'd need a bigger paper."
Mama scolds Papa for letting Fievel fall asleep with his face in a bowl of soup.
Papa: I figured he could eat a little, sleep a little...