Vash: I am known as Valentinez Alkalinella Xifax Sicidabohertz Gumbigobilla Blue Stradivari Talentrent Pierre Andrei Charton-Haymoss Ivanovicci Baldeus George Doitzel Kaiser III! Don't hesitate to call.
The episode with the drunken gunsmith. The end with Vash and the barkeeper.
Vash: AHAHAHAHA! HEY GRANDMA WHATCHA WAITIN FOR, FILL ER UP!
(Barkeeper bonks him over the head with a beer pitcher)
Vash: I mean please fill er up miss, if you wouldn't mind.
Barkeeper: That's better.
Earlier in that same episode:
Vash (who's drunk off his gourd): Gimme...gimme back mah gun grandma...(gets hit in the face with his gun) * THWACK!*
Frank Marlon: What happened, brother?
Vash: I dunno, my head just started to hurt.
And he still catches his gun!
Another episode, the famous "Genocide Song":
Vash: (singing) Total slaughter... total slaughter... I won't leave a single man alive... la dee da dee daa... genocide... la dee da dee daa... an ocean of blood... let's begin the killing time.
Thug 1: He's... singing?
Thug 2: What horrible lyrics!
Big Bad: Stay right where you are. He's just trying to scare you.
Vash: (sighs) I knew it wouldn't work.
And another episode:
Vash: THE DEADLY DODGEBALL HEAD! A simple technique to hold the ball in place with INTENSE SUCTION! Try this at home, kids!
Which brings in mind the first episode's Establishing Character Moment: the bad guy of the week is armed with a big-ass boomerang. Vash leaps over the weapon and sabotages the retraction mechanism so that it will wrap around the bad guy and squeeze him into submission.
Vash: (makes a cat-face) Dangerous toys are fun but you could get hurt!
In episode 5, "Hard Puncher", Vash is seemingly taken down, complete with blood on the floor. The looks on the mercenaries' faces when he calmly plugs the gun of one with his finger before appearing behind the rest with a jovial smile is absolutely hilarious.
And immediately topped by him nailing each of them twice with a child's dart-gun when he's threatened... before saying "It's okay, shoot if you want - try your luck!" Then he reveals that he'd counted the shots during their initial Blast Out.
(Meryl suddenly grabs her chest, as though she just got shot, and lets out an extremely melodramatic scream)
Most of the bar scene in Badlands Rumble, especially the part where Vash manages to steal the bullets out of everyone's guns without them noticing, and drunk Meryl.
Wolfwood killing Leonof the Puppetmaster shouldn't be hilarious. But his yell of "YOU LUNATIC!" and the explosion knocking him off his perch is, if unintentionally, hilarious.
During the first few episodes, Vash makes this exaggerated bishōnen face whenever he's around a girl he finds attractive. It wouldn't be that funny, except for the fact that it's so ridiculously over the top and comes out of literally nowhere.