A small double entendre, but in the book, the second letter is addressed to "The Smallest Bedroom." What is Harry doing at the start of the next chapter? Measuring his bedroom in steps!
"Uncle Vernon made another funny noise, like a mouse being trodden on."
Harry, on the "last day of August" (31 August) finally bites the bullet and asks Uncle Vernon if they can take him to King's Cross the next day so he can catch his train to Hogwarts. Vernon begrudgingly agrees, citing that they had to go to London anyway to get "(Dudley's) ruddy tail removed before he goes to Smeltings." Now, keep in mind when Dudley got that tail in the first place - when Hagrid first appeared and Vernon made the mistake of pushing Hagrid's Berserk Button. All this was back on Harry's eleventh birthday - 31 July - which means that at this point Dudley's been walking around with a pig's tail growing out of his bum for a month.
From the Stephen Fry audiobook, the sound he makes when Harry steps on Vernon's face as he's camping by the front door.
After Ron meets Hermione for the first time, he comments, "Whatever House I'm in, I hope she's not in it." Of course, not only does Hermione end up in Gryffindor with him, but the epilogue in the seventh book reveals that she eventually becomes his wife, meaning that Ron will live in the same house as Hermione for the rest of their lives!
It's a small one, but nonetheless the scene where Hagrid is in Gringotts and reaches into his pocket to find Harry's key only to come up with a bunch of dog biscuits counts.
Made even funnier when you realise that those biscuits are meant for Fluffy; i.e., a giant three-headed monster dog.
The Weasley Twins at the beginning.
"Don't cry, Ginny. We'll send you loads of owls.—"
"We'll send you a Hogwarts toilet seat!"
"Only joking, Mum."
Even funnier is that it's a callback to an earlier funny moment. Mrs. Weasley makes it clear that she wants the twins to behave this year, leading to her warning them "If I get one owl that you've... blown up a toilet or something..." This cues them to respond that they've never done that before, and are grateful for the suggestion.
Dumbledore: "I believe your friends Misters Fred and George Weasley were responsible for trying to send you a lavatory seat. No doubt they thought it would amuse you. Madam Pomfrey, however, thought it might not be very hygienic, and confiscated it."
He originally lost his taste for them after an unfortunate encounter with a vomit-flavored one.
After Hagrid tells Harry and Hermione that he's given Norbert a teddy bear:
From inside the crate came ripping noises that sounded to Harry as though the teddy was having his head torn off.
Realizing that in the first book, Fred and George jinxed snowballs to hit Quirrell's turban, and that they were unknowingly smacking Voldemort in the face!! You can just imagine Voldemort under the turban going "don't blow your cover, don't blow your cover, don't blow your—" *SMACK*
Dumbledore's Chocolate Frog card: "Professor Dumbledore enjoys chamber music and ten pin bowling." Best. Mental. Image. Ever.
Jim Dale singing "Hoggy Warty Hogwarts" for the audiobook of Philosopher's Stone.
"While he drove, Uncle Vernon complained to Aunt Petunia. He liked to complain about things: people at work, Harry, the council, Harry, the bank, and Harry were just a few of his favorite subjects. This morning, it was motorcycles."
"MOTORCYCLES DON'T FLY!"
When Harry gets the idea to have Charlie Weasley take Norbert off of their hands, he first launches the idea by glancing at Ron and muttering "Charlie". Ron's response?
"Now you're losing it too. I'm Ron, remember?"
Harry tells Ron and Hermione that Snape is going to be the referee for the following match.
Earlier, after Snape docks the trio points for a rule he basically made up, Harry notices that Snape is limping and wonders why. Ron's response is a bitter, "I don't know, but I hope it's really hurting him."
When we first meet the centaurs, they all comment that "Mars is bright, tonight" with Hagrid getting more and more annoyed.
"Mars is bright tonight."
Doubles as Hilarious in Hindsight, but early in the book when Vernon is intimidated by Hagrid he whimpers something that sounds like "Mimblewimble". Guess what? It's an actual spell.
Towards the beginning, Dudley tells Harry that they stuff people's heads down the toilet on the first day of Stonewall High and asks him if he wants to practice. Harry's response is 'No thanks, the poor toilet's never had anything as horrible as your head down it - it might be sick," and then runs away before Dudley can work out what he said.
A small one from Dudley, regarding the Hogwarts letters (also notable as it's probably his most civil moment with Harry in the first book):
"Who on Earth wants to talk to you this badly?"
Vocabulary lessons with Hagrid:
Harry: I never know, what's the difference between a stalagmite and a stalactite?
Hagrid: "Stalagmite's" got an 'M' in it.
From the film:
When Hagrid knocks the door off its hinges: "Sorry 'bout that!"
And he does it in the most casual tone possible.
Also when he's accidentally telling them loads of information about the Stone. "I shouldn'ta told you that! I should not have told you that!"
"Troll in the dungeon!" scene. Especially the terror-stricken look on Malfoy's face.
Emma Watson's delivery of the "killed, or worse, expelled" line.
Hermione during the flying lesson, when Harry persists in chasing after Malfoy: "What. An. Idiot."
Hermione's reaction when Harry says "Up" and his broom jumps into his hand immediately.
When Dumbledore announces that the third-floor corridor is out-of-bounds to anyone who "does not wish to die a most painful death" (and does so in the same businesslike way he handles all of the other announcements), the other students whisper amongst each other in curiosity. Meanwhile, Harry's face is pure WTF. It's #1 on this list.
Before McGonagall can even place the Sorting Hat on Malfoy's head, the hat shouts "SLYTHERIN!" with no hesitation.
In his first Potions class, Harry dutifully takes notes on Snape's introductory spiel: