All of "Party All The Time". Everyone's Played for Comedy attributes are employed...Shake is willing to destroy the sun if Frylock is cured of his cancer. Meatwad's line "Well, you should tell them that you don't have cancer" is fucking brutal.
Shake, Meatwad, and Carl trying to console one another after Frylock is diagnosed with cancer. Sure, a lot of the episode has the show's typical Black Humor, but... hell, even Shake is worried about him. Shake, Carl, and Meatwad all treat things as though they aren't protected by series-induced immortality. And by this point in the franchise, Frylock has never died. Not once. He's effectively the only character whose death could actually be permanent. The second episode of Aqua Unit Patrol Squad changed that, but the point stands.
The ending to "Mail-Order Bride", where Meatwad gives a heartbroken Carl a stick as a Christmas present. It obviously doesn't make Carl feel any better, but it's still pretty sweet of Meatwad to try and cheer him up.
A big YMMV: in the movie Shake kills Meatwad out of pure craziness but his reaction a few seconds later is pretty much one of the few times, if the only time, we see him genuinely remorseful for his actions, showing some degree of concern for Meatwad.
It's actually very revealing: Shake believes that Meatwad is invincible, and therefore, he can harm him with impunity. When it appears otherwise, though...
In A PE Christmas, Meatwad is given the chance to sing "Silent Night" for Shake's Christmas-themed rap album. It's actually done extremely sweetly, even if Master Shake's eel-induced diarrhea is loudly heard in the background. Better yet, Flavor Flav likes it! Bonus points go to Michael the sound board technician, who records Meatwad's song even though Master Shake ordered him not to.
A lot of Frylock's interactions with Meatwad are this in a way. He comes across as something of a father or older brother figure.
The end of the series finale: After over a decade of abuse from Master Shake and overall being the Butt Monkey of the trio, Meatwad gets married, starts a family, and overall finally leads a happy, normal life.Earn Your Happy Ending, indeed.
The entire series finale bounced between this and massive tearjerker. Special notices for when Carl actually helps out and shows remorse when Frylock dies, Shake apologizes for all the abuse he gave Meatwad, admitting it was to help strengthen him, and how Shake and Meatwad did everything they could to save Frylock, especially with Shake sacrificing himself to distract the clams from Meatwad.
Although the actual finale ended with Shake killed, Frylock and Carl wrongfully put in prison and can't end their sentence because they're both now immortal (not to mention that Frylock became a white supremacist), and Meatwad now an old man, and it looks as though this is how the series is going to end, it's a little relieving to see that Shake, Meatwad, Frylock and Carl okay after they see the ending on TV. Really, it's just good to see that after all that, they're still fine.
In "Multiple Meat," Although it's brief until you find out why he's doing so, seeing Carl playing Poker with the Meatwads is rather sweet.
A single line from Master Shake in season five's Hoppy Bunny. "I love family night." Seeing the usually confrontational and abusive Shake peacefully playing a board game with the rest of the ATHF, and actually expressing happiness at such a simple thing is oddly heartwarming.
In "Hospice", we see Carl chewing out his mother for the bad decisions she made that led to her awful health and trying not to breakdown while doing so. For all his berating, he really does love her deep down.
Then at the end of the episode, we see (via flashbacks) that Carl and his mom had a good relationship for most of his life.