Twice, Larry Talbot tries to warn Wilbur and Chick about monsters, including himself:
Talbot: Now, whenever the full moon rises I turn into a wolf. Wilbur: Oh, pal. That's all right. I'm sort of a wolf myself!
Talbot: You don't understand. Every night when the moon is full, I turn into a wolf! Wilbur: You and twenty million other guys. (Talbot tries to slap some sense into Wilbur)
One truly hilarious moment occurs when Talbot is untying Wilbur from the operating table Dracula tied him down to. Then he looks out the window to see the full moon. Wilbur nervously asks, "Now what, is someone else comin' after me?". The camera cuts back to show Talbot's suddenly hairy hands, and then pans up to show the Wolf Man in all his glory.
Chick: You're making enough noise to wake up the dead! Wilbur: I don't have to wake him up. He's up!
And soon after, when Dracula and the Monster are leaving McDougal's Wax Museum, they pass by a hypnotized-by-Dracula Wilbur. The Monster peers close and recoils with a cry of fear.
Dracula: Don't be afraid. He von't hurt you!
The sequence of Wilbur (Lou Costello) trying to sit down in a chair, only for a revolving wall to keep depositing the Monster behind him instead. They had to film multiple takes because Costello was improvising his reactions and the Monster's actor Glenn Strange kept breaking up in hysterics.
The scene where Chick and Wilbur are fleeing the Monster. They run into a room, push a heavy bed against the door, then lean their shoulders against it...and it turns out the door opens the other way. The scene is famous for delivering laughs and screams at the same time.
Lawrence Talbot, the world's most badass Butt-Monkey, while transformed into the Wolf Man, gets trapped under a tree. Wilbur assumes it's Chick in a costume, and proceeds to lambaste him for dressing up as a wolf after "that nice Mr. Talbot" asked him not to, and kicks him repeatedly.
The scene in the wax museum where Wilbur is reading the sign about Dracula's legend, only to be constantly interrupted by the opening of the lid on Dracula's casket. Wilbur's hysterical screams, followed by his lines with Chick, are sure laugh-getters.
Special mention goes to the very first reaction, as Wilbur pauses halfway through reading the sign and hears the coffin lid creaking shut. After pausing for a few seconds, he slowly puts the sign down, sighs to himself, calmly pushes himself up to his feet, and proceeds to scream at the top of his lungs.
Chick and Wilbur arrive at the castle to pick up their dates (both of whom are for Wilbur) and receive a call from Talbot, which Wilbur answers:
Talbot: "I believe you to be in the house of Dracula right now! You find the monster and I'll..." (sound of the other line being dropped) "Hello?" (cut to Wilbur trying to flee).
A comparatively subtle one: Bud and Lou are searching the castle and Lou finds a gothic cellar and quickly shuts the door. Bud asks him what he's found and he answers "Broom closet." In the immediately preceding scene, Bud had been examining the next door...a broom closet.
And in the very last scene, while Bud and Lou are escaping from the island, we hear these lines:
Wilbur: Well, he won't be bothering us any more. And another thing, Mister Chick Young, the next time I tell you I tell you that I saw something when I saw it, you believe me that I saw it!