The scene where the two TV personalities "Hard Gay" and Akihiro Miwa appear.
How about Taniguchi's expression when he saw Mikuru in a maid outfit.
Live Alive, the episode containing the former incidents and others, is made of this trope.
In Castle in the Sky, when the soldiers are in the glass-walled room that came out from the bottom of Laputa, you can notice Pazu peeking through the window from behind.
Miyazaki loves this trope in general as it enables him to showcase his characteristic attention to detail. Watch Calcifer eating up empty eggshells while Howl is cooking and Hiin scrambling to his feet in the palace stairs scene of Howl's Moving Castle.
In Lucky Star in the valentines day episode, a male student (Shiraishi, in fact) is seen giving Valentine's Day chocolate to another male student during a generic "front of school gates" establishing shot. Later in class, the two of them are seen in the background during a conversation by the main characters (who don't notice them), looking embarrassed.
The anime comedy Haré+Guu features this as a signature detail. Done by at least one or several groups of pokute and manda (the show's own flora and fauna), birds, and always during scenes in Guu's stomach. They mostly just mimic whatever the central characters are doing in a scene, but often they go off on their own scenerios. At one point, a bunch of pokute have a big laugh during a wide shot, and one of them actually rubs itself underneath Guu's skirt!! It takes at least a second viewing of an episode to catch everything.
Pokémon: After a brief shot of Pikachu activating a treadmill in the episode, "The School of Hard Knocks", we cut back to the main characters talking while Pikachu vs. The Treadmill carries on in the background.
In "It's Still Rocket Roll to Me", the gang accidentally get caught in zero gravity. While exposition was going on, Pikachu can be seen minding his own business swimming in zero-G!
In "Uncrushing Defeat", while Team Rocket have a lengthy discussion on the events of the previous episode, their Pokémon are playing with a roller the whole time.
BW42 takes the cake here, with so many things happening at once in a single scene: aspiring filmmaker Luke narrating his documentary at the foreground, while behind him we clearly see Ash and Stephan having an eating competition with Cilan watching in disbelief, and Iris and Bianca having a tug-of-war over Axew (which Bianca believes to be a disguised Zorua).
Ghost in the Shell: Innocence has a scene in a police station, with an incredibly detailed background with hundreds of people, including a man randomly riding a unicycle, and getting grabbed on all sides by cops. It looks like a scene from a slapstick comedy taken entirely out of context.
When first trying to play baseball, Koume from Taishou Yakyuu Musume does a somersault and ends up on her butt trying to catch a low ball. All this go unnoticed by the three characters having a conversation in the foreground, but not by the fandom.
Ken Akamatsu has a habit of throwing funny stuff in the background, though it's usually a shout out and not an actual joke.
Another one has Godel and Takamichiin a choke hold while everyone else discusses how Negi's power-ups have made him borderline non-human.
In Baka to Test to Shoukanjuu, while the members of the main Love Triangle are discussing where to go on a date-like... thing there is a fountain in the background shaped like a cupid that spouts water out of its butt, in varying volumes and pressures. Also someone rides by on a unicycle.
Also, in that very same episode while you see the main characters talk from an above angle, a shadow of a bird flies by, and then a shadow of a dragon goes by too. Other funny background things can be spotted too.
Sylphy from Amagi Brilliant Park is practically a walking Funny Background Event. In almost every scene featuring the four Elementario fairies, she's in the background goofing off while the other three are discussing something important to the plot.
In the Baratie arc of One Piece we see a man eating off a girl's plate. Word of God said that he was the girl's father and is indeed mooching off a bite from his daughter.
The running Panda Man gag. An OC of the author's who pops up frequently in the anime and manga, doing something or other in the background. Watch them all and an intricate story becomes clear.
In the Jaya arc, once Robin gets back to the ship, as Nami, who is still angry about what happened in Mock Town, blames Robin's claiming that there might be a sky island, Chopper uses Jumping Point, accidentally falls overboard, and Zoro goes after him.
Recently, when the Straw Hats are sailing to Dressrosa, a funny-looking, smiling crab - which has beome a Memetic Bystander in the meantime - appears randomly in the background while the characters are talking. A few pages later, you can see the ship's aquarium in the background, now the crab in it, which means that it has been caught and put in there off-panel.
I forget exactly when it happened, but there was a scene in an anime filler arc where Luffy nearly drowned and the rest of the crew had to pull him out of the water. As other characters were conversing, you could see Nami in the background slapping the hell out of Luffy in an attempt to "revive" him.
Done a few times in Keroro Gunsou. In one manga chapter, Momoka is gushing about Fuyuki while in the background, Tamama shoots himself with his own energy blast as part of his Training from Hell.
Fullmetal Alchemist is peppered with these. In one episode of Brotherhood, Scar causes a wall to explode, and for a split second you can see a startled cat flying through the air. There's actually a subtle (or not-so-subtle) cat gag every five seconds if you read the manga closely, usually involving Al.
At one point in Dragon Ball Z, King Kai is talking telepathically to the heroes and looking up Planet Namek. As he's doing that, his pet monkey Bubbles randomly starts hopping into the frame holding flowers for no reason—until King Kai whacks him with a hammer without comment or looking away from the book.
In the DBZ manga, the beginning of the Namek arc, Mr. Popo is standing on a flying carpet outside the window of 20th or so floor in a hospital. One floor below, an elderly man is staring at him and praying.
Yu-Gi-Oh! GX: After Daitokuji-sensei dies as Amnael, Judai, Sho, and Hayato have a conversation while Pharaoh, Daitokuji's cat, plays with his owner's soul, then swallows it.
Several times in the Crashtown arc of Yu-Gi-Oh! 5Ds, which takes place in an Old West setting, there's a place called "The Classy Ass" seen in the background in scenes that take place in the town street. (Given the nature of the town it's not surprising that there could be a business in Crashtown with a name like that. It was, of course, edited out in the dub.)
Yu Gi Oh ZEXAL has Yuma's and Kotori's grandmothers talking about their respective grandkids, while a group of old ladies swoon over the waiter, who is then challenged to a duel by an old man. (What makes this even funnier is that the duel is won by one of them using Tongue Twister, a weak monster that is not meant to be used offensively.)
A later episode has Yuma talking to his former opponent while Astral is checking out the construction site n the background. He looks rather amused.
During a scavenger hunt in a sports fest episode of Azumanga Daioh, you'll see Kagura frantically running around in the background shouting 'SCHOOL SWIMSUIT!' while Osaka borrows Yomi's glasses.
In the AzuDai manga, there's a strip where Osaka goes to ask Tomo about something — and we notice, but she apparently doesn't, that Tomo's got Chiyo-chan by the pigtails at the time. As they talk, we see Chiyo get released and run off.
In the Hetalia Bloodbath 2010, as Iceland was talking to Turkey about how he was getting an ominous feeling about how something bad was going to happen, in the background we see that something attacking the rest of the Nordics and kidnapping them.
Later, when the Nations are informed that they must figure out which of them has a mark on either their chest or their butt, England is examining his own ass in front of a full-length mirror in his office. As he reports in that the mark is not there, his assistant can be seen standing in the doorway, dropping his paperwork in shock.
Fushigi Yuugi gives us one example of this during the Suzaku Warriors' journey to Hokkan; Nuriko had just risked his life to save Miaka and Tamahome from being drifted off into a raging sea, all the while telling them something very profound. Miaka and Nuriko have a serious moment as they recall it. Meanwhile, Tamahome and Tasuki prepare to rip each other's heads off behind Nuriko.
Student council meetings take place on a balcony over the school, and only student council members are allowed on the balcony. Yet somehow meetings take place in the middle of a baseball game or just as a train passes by.
In other locales, Chuchu provides entertainment by eating things, causing trouble, and getting sucked into the occasional vacuum cleaner.
The duels are initially fought in an empty arena with two duelists and the rose bride. Later, the arena gets more crowded — an array of desks each with a milkshake which Anthy drinks one by one, or a collection of red sports cars sticking up out of the ground.
Mawaru-Penguindrum is absolutely made of this trope. Every single episode has the three penguins invoke this trope multiple times. (This should come as no surprise, considering that it has the same director as Revolutionary Girl Utena.)
In Bleach, Aizen is explaining to Isshin what is going on with the MacGuffin. While Ichigo and Gin are fighting it out in the background. Hilarity Ensues.
Also invoked in one of the endings, where Don Kanonji is flying off like a superhero in the background.
In one scene in the manga Goth, while Itsuki and Morino are talking about a murderer's notebook Morino found, a couple of their classmates can be seen respectively play-fighting and sprouting flowers from their heads in the background.
All Stars New Stage adds in another one. As Cure Sunny goes to talk to Ayumi, Cure Peace - who'd just got through talk to Ayumi - is snatched away by parts of Fusion. As Cure March steps in to nag at Cure Sunny, Cure Peace is tossed around, saved at the last minute by Cure Melody. The next scene shows Peace bowing profusely at Melody as Cure Muse lures the creatures away.
This scene in Sekai-ichi Hatsukoi. Watch everyone else in the background panic as Ritsu is trying to talk on the phone. Takano is screaming at Ritsu for the manuscript, Kisa loses the last of his sanity, Mino simply smiles but is obviously just as freaked out as everyone else, and Hatori...drops dead.
Urusei Yatsura: In the first movie, Ataru is running down a church aisle towards Lum with a ball and chain around his ankle. The ball knocks a guest out cold.
Detective Conan: While watching a movie, Ayumi and Ai fall asleep on Conan's shoulders, leaving him trapped for a page. When something happens behind him, however, he gets up to look. While the actions that will lead to Conan discovering the murder are occurring, a thud sound is displayed and the two girls are shown rubbing their heads while Conan tries to figure out what's going on.
Project Ako Final: scene in a bathroom discussing Miss Ayumi's marriage, behind the blue haired girl, a brunette is using a female urinal for the entire conversation.
In an early season three episode of Slayers, there's a scene where Xelloss is watching the antics of the other cast members from a rooftop and ends up laughing so hard that he falls off.
Tokyo Majin has a few, many of them because of the main character. An example is a scene where two characters are having a serious discussion on whether or not to join forces against demons at night while the main character slurps up his ramen... and chokes.
This◊ Air Gear page has a inverted example, while in the background the girls are commenting on Team Kogarasumaru's improvement in the running time, in the foreground, Buccha is holding Ikki in a Cobra Hold while Kazu forces his mouth open and forces Ikki to drink a deadly Gargle Blaster.
In Eyeshield 21, it's pretty hard to focus on Sena being happy about the Devilbats finally being able to play with a full roster when Monta is getting mauled by Cerberos in the background.
In the Persona4, episode 12: In Void Quest, while the Investigation Team is discussing that the killer might be inside the TV world in the foreground, Teddie is busy fighting off some dice Shadows with his Persona - and he's not doing a very good job of it.
When Ryuko first calls out Satsuki, everybody stares at Ryuko in shock. A boy in the background puts his hand on a girl's butt and she doesn't seem to notice.
When Mako asks Gamagoori what kind of pajamas he wears, he reveals he sleeps in the nude. A girl in the background blushes and puts her hand to her mouth.
In episode 19, Sukuyo spends the entirety of her screen time during an otherwise completely serious post-battle discussion staring at Aikuro's crotch.
In episode 9 of Gochuumon wa Usagi desu ka?, after Sharo snaps out of her depression in Chiya's tea shop, Anko, the latter's pet rabbit, starts to fall off Sharo's head and clings onto her shoulder/arm. It then grabs hold of her hand and starts to nibble on it while she's talking to Rize and Chiya. Made more hilarious by the fact that Sharo normally fears rabbits, but completely ignores it during this scene.