Right off the bat—poor dude is having his head forcefully flushed through his toilet at the behest of two random thugs...
Goon: WHERE'S THE MONEY, LEBOWSKI?! (shoves Dude's head in the toilet again) WHERE'S THE MONEY, SHITHEAD?! Dude: (legitimately confused) It's...it's... it's down there somewhere; let me take another look.
It very slowly dawns on them that they don't have the right place. "Isn't this guy supposed to be a millionaire, or something?"
Goon: (discovers Dude's bowling ball) ...What the fuck is this? Dude: Obviously, you're not a golfer. (Goon drops the bowling ball, cracking one of the tiles in the floor)
Inexplicably, the goons have switched outfits with each other when they return to The Dude's house.
"YOU SEE WHAT HAPPENS LARRY!? YOU SEE WHAT HAPPENS LARRY!? DO YOU SEE WHAT HAPPENS, LARRY, WHEN YOU FUCK A STRANGER IN THE ASS!?!"
The police chief in Malibu (combined with Nausea Fuel, because he makes you so sick) throwing the coffee mug right in Dude's face after he replies "Sorry, I wasn't listening" to his long and cautionary "The Reason You Suck" Speech (in a way, the irritant indeference makes it look even funnier while almost justified, nevertheless). Enhanced by Dude's reaction ("Ow, fucking fascist!").
As The Dude is being hustled into Mr. Lebowski's limo, he yells out, "Hey, careful, man, there's a beverage here, man!"
"Also, Dude, Chinaman is not the preferred nomenclature. Asian-American, please."
The thug who defaces the rug pronouncing "Ever thus to deadbeats" in the process, as if making a political point a la "Syc Semper Tyrannis" in the assasinations of Gaius Julius Caesar and Abraham Lincoln.
The Dude trying to pull a Royal "We" with Big Lebowski when he realizes he inadvertedly mentioned Walter came along.
The Dude thinking the boys in the picture are Big Lebowksi's literal children from different mothers,
Racially he's pretty cool huh?
Many of the gratuitous echoes and words one character borrows from another, "parlance of our times", unchecked aggression which will not stand, coitus, johnson etc...
The Dude, in good ol' Philip Marlowe mode, shades a notepad with a pencil to highlight the impression made on the next sheet of paper, only to discover Treehorn just drew a man with a big phallus.
Made doubly hilarious by the fact that the man is shaped like a giant phallus himself from torso upwards. In other words, a giant anthromorphic penis with a long erect penis of his own.
If you're mostly familiar with Philip Seymour Hoffman's serious roles, every one of his lines as The Comically SeriousLarge Ham Brandt becomes hysterical. Special mention goes to the limo scene:
Lebowski: They did not receive the money, you nitwit! They did not receive the money! Her life was in your hands!
Brandt:(adjusts glasses) This is our concern, Dude.
"But I hate the fucking Eagles, man..." Followed by the angry cab driver kicking the Dude out of his car.
This line by the Dude:
"Let me explain something to you. Um, I am not "Mr. Lebowski". You're Mr. Lebowski. I'm the Dude. So that's what you call me. You know, that or, uh, His Dudeness, or uh, Duder, or El Duderino if you're not into the whole brevity thing."