Emmy pointing out Descole's penchant for destruction and dress up. While he's standing RIGHT NEXT TO THEM!
Talking to Dean Delmona after solving his puzzles in Chapters 3-4 produces this gem that calls back to Last Specter.
Delmona: "Hershel, I expect to see a presentation on all your finds. No "a giant mechanical spider broke it" this time. Understand?"
Layton (looking ashamed): "Ah, yes. Of course."
Luke dressed up as a mouse, and Layton as a duck. To try and make Phong Gi's elder laugh. The clincher, though, is this: Picture Layton saying the following in the most deadpan tone possible:
Layton: "Quack quack".
Also the fact that, once the short cutscene is over, Luke's taken his mask off... but Layton is still wearing the duck face.
And Sycamore's attempt at a joke afterwards is so unfunny the chief doesn't even realize that was supposed to be a joke.
Even more hilarious is how huffily indignant Sycamore gets when the joke flops, blaming it on the natives not being sophisticated enough to get it. The pissiness becomes even more hilarious when you get the reveal that he's Descole, the genius antagonist of the prequel games.
Better still when you go back for a World Time's article, and Professor Sycamore is encouraged to tell another joke. To stop the chief from laughing.
The usual mirror gag:
Luke: "Who's that in the mirror? Why it's Professor Layton's star apprentice!"
Emmy: "Oh! Is that Professor Layton's star assistant I see in the mirror?"
Layton: "Good day to you, fine gentleman. We truly must stop meeting like this."
Later on, if you go back, Sycamore and Aurora join in the fun as well.
Luke eying up a leg of ham:
Luke: "Oh, look! It's a leg of ham!"
Luke: "That ham looks really tasty..."
Luke: "Mmmmmm... Haaam..."
Luke also touches a flame while at the Grand Stage in the Jungle, reasoning that, "they say curiosity killed the cat. I don't see any cats so... OUCH! I guess that's purr for the course..."
One of Descole's "puzzle solved" speeches!
Descole: "Thank you, thank you. I'm here all week!"
Descole's "incorrect guess" speeches.
Descole: "No! That's not what I meant to say!"
Descole: "This puzzle is clearly flawed."
Mentioning who says this line is a spoiler, but:
"Hmm. You know, this construction reminds me of a... painting."
Poor Professor Sycamore just can't get a break in the game, between people making fun of his hairstyle and jokes.
Amanita: It's... your hair! What's with those funny little bunches? They're so curly, like little piggy tails!
Sycamore looks both horrified and offended by this.
It becomes even more Hilarious in Hindsight when it's revealed that he's Descole, Layton's archnemesis of the prequel trilogy and a usually suave villain who pulls off all sorts of stunts.
And if you tap the snowman beside the cave in Froenborg after Sycamore is revealed to be Descole, Luke makes this remark:
Luke: Did Descole really stand here building a snowman?
If you go back to the snowman after Descole joins your party, here's what he says:
Descole: "A finely sculpted snowman, if I do say so myself."
After Inspector Grosky almost gets beat up by a gang of Targent thugs when Emmy saves him, he declares pompously, "Another win for the yard!"
Quite a few of the run-ins with Targent agents, most notably Robin and Macaw, are pretty hilarious. Some notable instances:
When Robin and Macaw chase the group to the island, they are utterly confused by all the Ponopos for sale and have no clue how to find the egg they're looking for. They try to resolve the issue by buying as many as they can carry. Then it turns out that they were offered the real egg as a part of the Harmony Ring but refused it, thinking that it was a fake.
At one point, Macaw and Robin get it in their heads that the egg they're searching for is being carried by a vulture Luke had previously been talking to. They run of leaving the group to wonder if they seriously intend to make the vulture lay an egg for them. Luke is hilariously genuinely concerned, as he's fairly certain the vulture was male.
In one of the bonus episodes, we see a Targent agent from Phong Gi. He's dressed in the same snappy suit and sunglasses as the rest... but also has an enormous afro. And he turns out to be the nephew of the native who kept laughing at Sycamore's hair.