The ending to "Dog Tags" is surprisingly sweet for this show; when Ren can't gain access to his own dog club, Stimpy gets the idea to bring him to a club for cats. Ren (dressed as a cat) and Stimpy sing "When Irish Eyes Are Smiling" with some other cats and then share a bowl of milk together. Awwwww.
Pilot episode "Big House Blues". Ren and Stimpy get put into shelter by a dog catcher and a little girl chooses Ren for her pet. After he's taken out of the cage, he mumbles at the girl "If you take me, you have to take Stimpy too!".
"Stimpy's Fan Club". Ren is overjoyed to finally get a fan letter for him, after a stream of fan mail for Stimpy. He gloats about it and shows it off... only to find out the letter was from Stimpy himself.
Stimpy: I meant every word.
Ren (bursting out crying and snuggling up to Stimpy): Look at me! Big, mean Ren, crying like a baby! Pretty pathethic, huh?!
Stimpy: Oh, peeshaw! No one will ever know. You just cry your little eyes out.
"Son of Stimpy". Crowning Moment of Heartwarming combined with a Crowning Moment of Funny, since the whole episode is dedicated to Stimpy's fart, that he happens to believe is his son (originally aptly titled "Stimpy's First Fart"). Stinky (as in, the fart) leaves his home (as in, Stimpy's butt). The whole episode is spent on Stimpy trying to find Stinky, and is sealed with a heartwarming reunion.
Don't forget Ren. His grief at losing Stimpy and his joy at having him back are downright beautiful, especially considering it's Ren we're talking about.
The end of Stimpy's bedtime story "The Littlest Giant". The eponymous giant saves a village from a terrible drought. In return, he gets what he always wanted: a friend.