Animaniacs has a high list of running gags. Most of them are catchphrases.
Yakko, Wakko and Dot
Dot's Pet occurs many times throughout Yakko, Wakko, and Dot segments. Upon the attempt of any monster to scare Dot, Dot will take out a small white or pink box, claim it to be her pet, and open to reveal another monster. From this point the first monster will either run away or fall in love with Dot's pet.
The first running gag ever seen was the Hello Nurse exclamation, also in the Yakko, Wakko and Dot segments. Enter a pretty character of the opposite sex, and the respective Warner sibling(s) (or Slappy, in one cartoon) will exclaim, "Helloooooo Nurse!" This is one of two extremely extremely common gags that everyone ever watching Animaniacs has seen, so much that on occasion they say or reference it multiple times an episode.
Another gag commonly seen with Yakko, Wakko and Dot is when Ralph chases them across certain other segments.
A very famous running gag is with Yakko blowing a kiss and saying "Goodnight, Everybody!" to the audience after a Hidden Innuendo is made.
For almost every episode, the Godpigeon will walk up to the Goodfeathers (or sometimes just one Goodfeather) and mumble in a way only Bobby and the Narrator can understand correctly. Also, when the Godpigeon walks up to the Goodfeathers (or one of the Goodfeathers), he has them kiss his toes.
The other principal gag in Goodfeathers segments is Squit complimenting Pesto, and Pesto interpreting it in a ridiculous way.
Squit: Wow, Pesto, you're really ______.
Pesto: I am ______? Are you saying that I am some ____________? Am I a ____________, here to amuse you?
Squit: No, Pesto, I was just saying that you're ______, that's all!
Pesto: ...I am ______.
Squit: Yeah, yeah, that's what I'm saying!
Pesto: ... THAT'S IT! *starts to beat up Squit*
Bobby's reaction to this is either telling Pesto to lay off Squit or laughing his head off.
Buttons and Mindy
Every episode has the same premise. Near the end, or the middle, Mindy will run across either one person or 2 people, and pester them with questions of "Why?" until they run out of answers, at which point Mindy will say "Okay I love you bu-bye!" and continue.
Pinky and the Brain
Every Pinky and the Brain segment on Animaniacs has Brain asking Pinky whether or not he was thinking the same thing Brain was thinking, and Pinky would always reply with a non-sequitur . On the Pinky and the Brain spinoffs, there were two where Pinky was indeed thinking what Brain was, one where Snowball answered Brain, and one where Brain answered Pinky.
In certain Pinky and the Brain segments, Brain would hit Pinky because he did not refer to Brain by a certain title appropriate to the episode.
In most episodes where Slappy Squirrel appeared, Slappy Squirrel would blow someone up, sometimes going to great lengths to set the explosives up. Normally, she also follows up with, "Now that's comedy". Skippy appears to share this affinity for blowing people up.
In the Wheel of Morality segment, Yakko would spin a wheel, and it would land on a number, which would cause a small printer beside the wheel to print out the moral. Then Ralph would catch up with them and they would run away. He takes noticeably longer when Wakko and Dot start whining that "the morals make no sense", until Yakko reminds them that Fox wanted it. If the wheel didn't land on a number, they would start cheering that they had "won".
In Rita and Runt segments, Rita would always sing, except for in Episode 35 when they were separated. Runt would always remark that she sang beautifully.
In Hip Hippo segments, Marita would often complain/worry about not being on the latest trends, and Flavio would assure her that they would catch up as soon as possible.
Certain episodes have gags associated mainly with them.
Episode 5: Taming Of The Screwy: Likely the biggest chase scene in the history of this show.
Episode 9: Wally Llama/Where Rodents Dare: Wally refuses to answer any questions, which could be considered a running gag but more likely just a premise of the episode. Also, in Where Rodents Dare, Pinky would start drumming, and Brain would tell him to stop because it was disrupting his thinking.
Episode 10: King Yakko: The most obvious running gag here is Dot showing everyone red uniforms with white polka dots all over. Someone would question the dress style by saying "Polka Dot?" and Dot would interpret this as them asking her to dance. Then either Yakko or Dot would start playing the accordion and the Warners would start dancing with people nearby.
Episode 12: Garage Sale of the Century/West Side Pigeons: Wakko managed to get ahold of Pa Bear's garage control, and started using it to move objects. This would prove instrumental to the story line when Wakko proved that Pa Bear was lying about having donated his profits to charity.
Episode 13: Hello Nice Warners/La Behemoth/Little Old Slappy from Pasadena: The narrator repeatedly tries to prove that Slappy is exceeding the speed limit.
Episode 17: Roll Over Beethoven/The Cat and the Fiddle: The Warners' incessant humming is as much of a running gag as Wally Llama's refusal to answer questions is.
Episode 26: Testimonials/Babblin' Bijou/Potty Emergency/Sir Yaksalot: All three testimonials agree that Milton Berle disliked the Warners, which could be a running gag; the narrator reminded the last 2 testifiers that it had been covered in the past. In Sir Yaksalot, there were three instances where a certain person or group of people shouted "The dragon!" repeatedly (first a charred man running into King Arthur's room, then King Arthur himself trying to persuade the Warners to take action, then the peasants at the defeat of the dragon), and each time, Yakko would ask for someone to stop them from yelling "dragon", and the writer would oblige by drawing an anvil over them.
Episode 40: Casablanca Opening/Fair Game/The Slapper/Puppet Rulers: The answer to every question is Isaac Newton. That show must realize the gravity of his accomplishments... *gets shot*
Episode 43: Survey Ladies/Of Nice and Men/What a Dump!: The Warners really really aren't interested in the survey ladies' George Wendt with beans survey, and they express this in different ways. Then again, it is the main idea of the cartoon.
Episode 46: Turkey Jerky/Wild Blue Yonder: The Warners possess a set of houses of varying size which they use to shrink Standish so that Mr Gobbles can crush him.