The horse chase scene. While Layton and Emmy get good horses, Luke gets a smaller one that looks pretty derpy with its tongue sticking out.
Luke's "horse" actually isn't a horse at all — judging by the tail and the length of the ears, it's a donkey. Which makes his comment ("Mine has something wrong with it!") all the more priceless.
Layton just enjoys recalling puzzles in the weirdest of ways.
Narcisse: Just look at my flawless nose, my glorious chin... my, aha, effortless smirk!
Layton: Hm, now that you mention it, your face does remind me of a puzzle.
Emmy: Careful, Professor! Not everything is a puzzle!
Layton: Oh... I'm sorry. I could have sworn there was a puzzle here. My mistake.
The English version of this scene has a somewhat different subtext. Layton's comments come off as if he's coolly contemplating rearranging the (extremely smug and condescending) guy's face. Emmy seems genuinely alarmed. It's priceless.
Luke Vs. the Cactus.
Luke: Okay, cactus, you win. For now...
During one conversation, Emmy expresses concern that the ferris wheel could come loose and start chasing them around the park. Luke's reaction is Hilarious in Hindsight if you've played the other games:
Luke: A ferris wheel chasing people around? That's just silly!
Emmy in the town hall:
Emmy: (Upon seeing comment cards) I'm tempted to write silly things on them!
Layton: I would advise agai— Oh, she's already started. *Pulls hat over his eyes*
When you examine the ticket booth in Tingly Town:
Layton: "Families get 30% off". Yes, yes, I know what you're thinking...
"While I do not enjoy fighting, I find the idea of a puzzle battle to be quite alluring. Or, to use a colloquialism: please bring it."
A partial one in the part in the post-credit cutscene where Descole punches out Bronev's mooks... accompanied by bouncy sound effects.
When examining the mirror in the hotel room:
Luke: Professor, look, it's your number one apprentice!
Emmy: Professor, look, it's your number one assistant!
Layton: Oh, look, it's a dashing gentleman in a top hat.
Pretty much everything Madelaine/Serena says about the Masked Gentleman.
The final gift for having completed all 150 puzzles of the game. It shows the group of friends all grown up (and Luke), with Randall trying to take Layton's hat off. Layton tries to keep him off from his hat.
"Mr. Layton! This certainly is a pleasant... intrusion."
"So now you're waltzing into people's rooms unannounced, then?"
Randall and his obsession with hiding puzzles around town.
Emmy and Luke's reactions to learning about how Layton used to keep his hair in an afro. Luke wonders if Layton used his hairstyle as a convenient means of storing stuff.
Emmy's reaction to finding out that really young-looking clown named Puck is actually an adult:
Emmy: Wait. What?!
Puck then follows up this little reveal by telling Luke:
Puck: Look kid, remember this: if you're cute and little, people will always be nice to you. Especially beautiful women.
A groan-worthy line from Layton on the roller coaster. For bonus points, he actually grips the brim of his hat as he says it.