The intro to the game is a Continuity Nod to that of the game before it, Phantom Hourglass. We start with a story being told through a slideshow, presented by Niko no less (and considering it's been over a hundred years since then, the guy REALLY hung in there). Link fell asleep through the story, which causes his master, Alfonzo, to wake him up by shaking him silly.
Chatting with some of the Phantoms in the Tower of Spirits can find some amusing results.
"YOU SEEM CUTER THAN USUAL. WHY."
"CAN'T YOU SEE I'M WORKING HERE."
"MY JOB IS TO SLOG THROUGH LAVA ALL DAY. BOO."
"MY TORCH IS THE LATEST IN TORCH TECHNOLOGY."
"THE PATH I'M WALKING HAS A SPECIAL MEANING."
"I WISH WE HAD A SPECIAL MOVE, TOO." "THE FOUR OF US SHOULD INVENT A SPECIAL MOVE."
It can be pretty funny watching the Wrecker Phantoms transform into Morph Balls.
After the Sand Temple, a cutscene ensues of Zelda and Link rushing towards Anjean excitedly, telling her what just happened. Zelda then asks how they find Malladus. A mere response of "I don't know" from Anjean causes Zelda to crash to the ground, which is pretty weird considering she's a ghost a this point.
When we find out that the demon king intends to possess Zelda's body, she reacts with appropriate horror, and goes on a mini rant about how Link has to get her body back now, while she waits here for him to get the job done. Family tradition, after all. But what really sells it is her eyes, deadly serious as she floats closer and closer to Link, and you just know that she would be shaking him by the shoulders if she could. One of the only few times where you can see Zelda with an angry face. Watch it here.
The moment she shows Link her ability to possess Phantoms. It creeps Link out and causes him to give a nervous chuckle, but it ends as a funny scene.
Also, it's inherently funny, seeing a big, intimidating, walking suit of armor like a Phantom, doing girly gestures and the like.
There's a sidequest where a woman in Papuchia Village is looking for a husband, but she's rather picky and the only suitable candidate is the Chief of Whittleton. However, she's pleased with the choice and they become a couple. Later on in the game, you rescue a prisoner from Pirates' Hideout and he says that he cannot wait to meet his girlfriend who is waiting for him... back in Papuchia Village. Even Zelda points out how awkward the situation became.
Spirit Tracks is the perfect parody of your typical Shonen Anime: After Link and Zelda defeat Byrne/Staven, who then insists that This Cannot Be! since they're just human, Zelda proceeds to give him the stereotypical "With The Power of Friendship" speech, followed by Bishie Sparkle radiating from Link's smiling face. They proceed to giggle at each other, give each other High Five and everything seems lucky and merry and happy... until the music suddenly dies down and they realize that Byrne, who's not quite finished off yet, has been ignoring them and took the exit in the back.
Made even better by the fact that during this scene Zelda is possessing the body of a phantom about 5 times the size of Link.
Zelda's boxing movements when saying that she wants to have words with Byrne after she gets her body back. Byrne's angry eye-open and glare off to the side only seal the moment.
The scene after Zelda regains her body mid-air and begins to fall. Byrne shouts "Go now! Right now!" and the heroic overworld music starts to play as Link runs forward to catch her...only to cut off abruptly as Zelda just falls and crushes him.
Going on a long quest to bring one of the Lokomos a massive, glittery vase... and then smashing it to pieces. His reaction is priceless, your reaction choices to his reaction are funny, and his reactions to both of those choices are downright hilarious.
Near the beginning of the game, after mopping the floor with Alfonzo, Byrne approaches Zelda in order to capture her. But wait! Link jumps in front of her to protect her from the evil man... only for Byrne to slap him several feet to the side, knocking him out instantly.
It's small, but during the confrontation between Byrne and Anjean, notice Link and Zelda's absolutely spooked faces when Byrne makes The Slow Walk right past them while ignoring the duo completely.