Situations that looked like they were going to cross into squick territory became CMOH that were both sweet and funny. The various hardened thugs, rapists and killers putting all their efforts into a children's birthday party was cute AND hilarious.
There's also the blink-and-miss cameo in "Mayhem Donor" where Ash and the ghost of the girl are hiding out in a burrow together far away from the chaos in the jail, with the little girl reading a storybook to him.
Jailbot's dream in "The Dream Machine", which consists of Jailbot as a human child (with his dot matrix face) and the Warden as his father fishing together. Even the Warden is touched by this moment, saying he knew Jailbot couldn't have been the one plotting the (non-existent) riot.
The moment in the season finale when the Twins reunite with Combaticus might qualify as well.
The first season finale where Jared and Alice find Jailbot, and upon freeing him, Jailbot gives the pair a hug and his facial display turns into a heart. It's only a moment, and is immediately followed by Jailbot grabbing a kid and shaking him inside out for the seagulls to eat.
Whenever Alice displays her secret desire to be a mom, with her caring for the little girl in "Mr. Grumpy-Pants" or believing that she'd be get to be the godmother of the Twins' baby.
Some also find the end of her origin to be an adorable moment: After Alice starts her sex change and winds up fired, she is extremely upset and feeling quite down (with her explaining in a voice-over that no one would hire her). The Warden comes out of nowhere and tells her she is beautiful—and he means it.
Jailbot setting Jackknife free at the end of the second season premiere.
There's a blink-and-you'll-miss-it one in "Ghosts": late in the episode, when the ghosts are invading Superjail, you briefly see Ash reunited with the ghost of Sanser. It's also subverted in the same episode when Jackknife sees his dead father again. First they hug—and then they start trying to kill each other.
In "Uh-oh It's Magic" Jared gains popularity as a stage magician and when he was fired from being the manager of the Warden's sentient ventriloquist dummy, the Warden actually treated him like a friend and said that he understood what fame does to a person. He and Jared even stop the prison escape in a combo of this and CMOA.
In "Oedipus Mess", Jacknife grabs a bunch of newborn babies and throws them behind him, having the pursuing Jailbot have to slow down to catch them. When Jailbot saves the final few, he leaves with Jacknife, showing that the hospital has a statue engraved with the words "Cancer Memorial Park" complete with a statue of a happy San-ser holding a slice of cake, wearing a party hat.
Within the same episode, the Twins have this line of dialogue summing up their relationship after being "dumped" by their Jacknife clone friends (who they weren't paying much attention to to begin with): "Us multikind are never truly by our lonesome." Of course, this is being said while the Time Mall's equivalent of Jailbot is busy slaughtering innocent aliens in an attempt to catch the Jacknifes.
The final scene in Oedipus Mess starts out seemingly adorable, but quickly moves into a darkly funny moment: Jacknife is awarded his son and left to care for him. However, he and his son both instead decide to dig their way out of the prison cell together, with the baby Jacknife giggling "DADA!" in response to his father's grunting.
The Warden winds up imprisoning Jean in a solitary confinement tower in "Sticky Discharge". When Paul decides he doesn't want to leave Superjail, he proceeds to fight his way back to the tower and survive the bloodbaths that kill the other inmates, then make it to Jean's cell.
Overlapping with a moment of awesome, there's the bit in "Terrorarium" where Jean and Paul are facing death from the giant bugs- only for Jean's "Strongest Cockroach" to save both of them and fly them off to safety. The cockroach is even depicted on the season 1 poster artwork, happily carrying its owners off from the brawl that's going on.
Not quite part of the show, but the music video for "Winner" has the Cheeseburger member Jayson Green telling Lord Stingray, Jared, Ash, and the Warden that they are winners and should be confident in themselves.
The Season Four premiere Superhell ends with Charise believing that Jared is dead and moving on by hooking up with the Doctor, making for a tremendous Tear Jerker. By the end of Superstorm, however, Jared and Charise are back together!