Any time your Pikmin tries to carry treasure by itself, or starts taking on a large enemy solo. Granted, it will usually get killed in the latter case, so this can be annoying. Especially funny if the Pikmin in question is a red leaf Pikmin carrying back a fiery blowhog.
The animated shorts
"The Night Juicer":
The red, yellow, and blue pikmin all think that the pikpik carrot Olimar just juiced up is one of their own, and are afraid that he's reaching for them next. Olimar hardly even acknowledges that they're there, even casually pushing the Red aside to grab more carrots from the box they just knocked over.
Similarly, a yellow pikmin gets the idea to use a Bomb Rock to blow up the bottle and get the red pikmin out. All the Pikmin in the general area react accordingly when they see him coming with a lit one held in his hands. However, the Bomb Rock idea actually does succeed in getting the red pikmin out by sending the bottle flying into the air and letting the Red reach the exit... however, no sooner does he start to celebrate does he land back in the bottle shortly after.
When the red pikmin is finally freed from the bottle by the other pikmin, it joyfully hugs the blue pikmin, before leaping towards the purple pikmin and hugging its face, nearly causing it to stumble off the bottle.
"Occupational Hazards":
There are a few funny ways which the pikmin carry back the parts of the digger. One red is rolling on top of another red, covered in washers. Another red is carrying two washers back by putting them around its arms and rolling them like wheels. At first it looks like one really tall Red pikmin is carrying six nuts at once by putting them around its body, but it's shortly revealed that it's three pikmin carrying two nuts each stacked on top of each other.
After getting kicked in the eye by a pikmin hiding in a wheel, the enraged bulborb tries to kick the wheel and remove the hiding place. Evidently though, bulborb legs aren't very strong (and besides that, the wheel is partially buried in the ground). The resulting hopping around the bulborb does on one foot is pretty hilarious.
When the yellow pikmin enters the cave with the bulborb in it while spotting some treasure, it eventually ends up waking the bulborb up in curiosity. When the bulborb turns around and spots it, despite Yellow's lack of facial features outside of its eyes and ears, it still manages to give a pretty epic Oh, Crap! expression.
Later on, when we're shown the scene with the yellows again, we see that tons more have joined since the above scene, their buds and flowers glowing different colors from the electricity as they all hold onto each other, strung about every which way. The end result ends up looking like Pikmin-themed Christmas lights.
Early on, the Pikmin come across some dusty pipes, going through them results in the Pikmin getting covered in the dust, one blue Pikmin gets a dusty afro. The best part however? A yellow pikmin pulls out a mushroom, and the camera pans up to a very familiar green pipe, coupled with gold coins popping out with the appropriate sound effects. then, the red Pikmin in said pipe gets dust all over its face that resembles Mario's hat, nose, and mustache... with the first few notes of the Super Mario Brothers theme playing as he jumps out with the pose and sound effect of Mario's jump.
As the Pikmin and the Bulborb are dumped from the digger's scoop, the Pikmin's leaves quickly sprout into flowers to allow them to safely glide down to the ground. The Bulborb gets an Imagine Spot of its mushroom pattern extending to act as a parachute... before realizing that it can't do thatand falling to its death.
When the Bulborb dies from its fall from the digger's scoop, many of the Pikmin start to gather around it. When Olimar realizes that the digger was collapsing, he calls back all the Pikmin from the Bulborb... and they leave behind a censored space. What did the Pikmin do to warrant that kind of censorship?!
Given the noises they're making are Pikmin Digging/Combat noises, it's very likely that Olimar's Pikmin were subjecting the Bulborb's corpse to Extreme Mêlée Revenge for the Pikmin it killed. Even funnier is that they do this without any prompting from Olimar.
The Pikmin also almost trample Olimar in their rush to the Bulborb's corpse, after which Olimar looks at their furious corpse dogpile and gives the camera a nervous Aside Glance complete with a bead of sweat rolling down his face.
The cost of the destroyed digger: 1,000,000,000,000,007 Pokos!note That's a literal quadrillion, for those curious Olimar makes a triumphant jump and pose, and you can tell: he's about to salvage that digger. Another interesting note is that, in Pikmin 2, Red Bulborbs (of whom one is trapped under the wreckage) are valued at seven pokos.
The digger collapsing itself is pretty hilarious since Olimar didn't stop to think about what removing all the bolts from a giant structure would do. Luckily for him, this also unintentionally made salvaging the incredibly valuable wreck easier for him.
A White Pikmin gets a nail dropped on its head and it starts angrily squeaking... then it sees two Blues trying and failing to launch a Purple Pikmin on a catapult-like contraption and just gets angrier. The Purple steps aside and lets the White go... only for two Purples to jump onto the other end of the catapult, sending the White Pikmin flying.
To top things off, as a Brick Joke, on the ending screen the white pikmin lands in front of the sprouts, then the bolt conks it on the head ticking it off again. You know, just in case anyone was wondering What Happened to the Mouse?.
