Godzilla mentioning he has a peanut allergy in the very first episode.
Rodan declaring he will cut his dick off after seeing Mothra dating King Ghidorah, claiming it will distract him from the pain.
The episode "Godzilla PI" centering about a private detective-themed novel which Rodan wrote has plenty of gems.
Zilla in his earlier appereances is a walking moment of funny. He is esentially a smaller, chubby version of Godzilla that talks with a British accent. To top it all, the toy used to represent him is locked in Godzilla's (in)famous dance which he performed after beating Ghidorah in Invasion of Astro-Monster.
Rodan staying behind in a Gamera movie just to watch the sex scenes and then being quickly pulled away by Godzilla.
Gigan's introduction and his attempt to enter Godzilla's band.
Godzilla and Rodan going completely drunk.
Jet Jaguar having sex with Hedorah. When he first sees her, he says, in a dumbfounded voice "She's beautiful". Hedorah herself speaking in subtitled farting noises all the time as well.
Rodan mistaking three ponies for King Ghidorah at a bar.
"Monsters in Training". As the upbeat, motivational song of the same name plays in the background, we are threated to shots of Godzilla and Rodan constantly failing at every exercise they try.
Jet Jaguar after Mechagodzilla rebuilds him. Particularly his original function, which was singing and playing the banjo at childrens birthday parties at Chuck E. Cheese's and his awfully long serial number.
King Ghidorah's final words before his first defeat:"Oh shit!"
The bit of Self-Deprecation in the season finale. After Godzilla (rethorically) asks "who writes this crap, anyway?", a live-action scene of the creator wearing a Godzilla mask shows up, laughs and then says "Ain't i a stinker?" while giving the thumbs up.
The formation of LESBIANS. Also, Earl for being the token human in a group of monsters. The LESBIANS laughing evilly at the end with Gigan thinking they're his friends is also a hilarious scene.