Cloudy with a Chance of Meatballs is rife with them. Animator David Gibson mentions on his blog that he got in a little trouble "because I thought it would be funny to make one of [the background characters] mouth (in slow motion) "What The F#* K?." Haha right? The producer didn't think it was funny." (It's still there, though).
During the climactic fight against the Omnidroid, the family gets a hold of Syndrome's remote. After the Omnidroid uses a Rocket Punch to grab Mr. Incredible...
Mr. Incredible:*gets tossed into the air by the claw opening* "Huuwaaugh!"
After Lucius comes and gets Mr. Incredible, Jack-Jack can be seen in the back trying to eat the spit that Lucius froze.
The argument between Lucius and Honey as the former tries to find his supersuit, with the chaos the Omnidroid is inflicting in the background.
The Lion King has Timon and Pumbaa changing poses every time they swing by during young Simba's solo in the Hakuna Matata song. The animators lamented that they spent all this effort on the background gag and no one noticed.
And then it's given a great deal of lampshading in The Lion King 1 1/2.
In the last part of Be Prepared while Scar is being a Large Ham and bragging about how he'll be king the hyenas are singing about how much food they'll get working for him.
In The Swan Princess, during the song "This Is My Idea" you see several of these; the most notable of which is when - as children - Odette trips Derek on the stairs, takes a flying leap on him, and they start beating the crap out of each other.
Surf's Up has characters doing funny stuff behind the interview scenes, like the penguins peeking in the window of Cody's mom, or the little penguin who keeps pretending to drown to get Lani's attention. Also, there are flying saucers hidden on some shots.
Adventures of Tintin has Snowy, where in the desert scene, he appears out of nowhere carrying a bone. Which is actually very important, seeing that it's a CAMEL's leg bone, which tells Tintin and the captain that they are somewhere in the Sahara instead of Spain as they'd planned.
After Thompson and Thompson pull the pickpocket out from the Circling Birdies moment, the old lady who the pickpocket crashed into strikes a man with her cane, thinking he was groping her.
In Aladdin, his monkey Abu harasses the owner of a melon stand. The scene is so distracting, many viewers fail to notice Aladdin's hands in the foreground taking a melon for himself.
Quick example in A Goofy Movie. At Lester's Swamp kiddie park, Max loses his temper with an annoying employee dressed in a Lester Possum suit and punches the guy in the face, turning the fake head backwards. As Max trudges gloomily towards the camera (away from the tourist trap), we see "Lester" fumbling in the background... until a group of small children knock him over and drag him offscreen.
Before that, Reno attacks Cloud at Shinra's house. Cloud responds by locking him outside, and for the rest of the scene, you can hear Reno asking for someone to let him back in.
Towards the end, Rude and Reno are attempting to climb a tarp to rescue Shinra. Reno climbs nimbly up (as seen in an earlier fight scene) while Rude loses his grip and falls to the ground, lying on his back and waving his limbs like a turtle.
When Rainbow Dash talks about her role in the Rainbooms in the band room, Pinkie Pie can be seen wearing a cymbal on her head like a hat.
During the "Let's Have a Battle (of the Bands)" song, most of the students are entranced by the Dazzlings... except DJ Pon-3 who's listening to her headphones and is oblivious to this. Becomes an important detail later, however...