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.
The 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 Thomson 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...
At the party in the gym, when Twilight and the Rainbooms hold hands to defeat the Dazzlings, Pinkie can be seen with a cookie in her mouth and her fist in the air. As the others wonder why their standing together in unity isn't unleashing magic, Pinkie is still seen with her fist in the air, the whole cookie stuffed in her mouth, and crumbs on her chin.
In The Book Of Life, as Joaquin digs himself deeper and deeper agreeing to Maria's "Women should be at their husbands beck and call" trap the shortest Nun can be seen gripping her knife and glaring at Joaquin.
After Lord Business threatens Bad Cop with the Band-Aid, he hands it off to two of his mooks and continues talking. They...have a bit of trouble handling it from there.
While Wyldstyle is talking to Vitruvius in the saloon, Emmet is seen interacting with things in the background. He even takes a sip of alcohol and gags.
When the team is driving up the rainbow to Cloud Cuckooland, two knights can be seen jousting, with one completely skewering the other.
Inverted with a Funny Foreground Event; during Wyldstyle's big speech to rally the people against Lord Business, everyone turns to the big TV screens... except one guy who's listening to his headphones and keeps walking until he's shot by a TAKOS.
When Batman tells Wyldstyle "He's the hero you deserve", gesturing to Emmet, Emmet looks around to see if Batman is gesturing to someone behind him, whilst in the background, Benny also looks around to see if he's referring to someone else.
During Bad Cop's interrogation of Emmet, one of the robots who's supposed to be watching the interrogation is instead playing Solitaire on his computer.