This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / Rocko's Modern Life
Tickled Pinky, the episode where Rocko has to have his appendix removed and has a dream before his surgery about spending one last day with and anthropomorphized version of it, a'la the "Make A Wish Foundation." It still seems to have been tailored to wring tears from anyone who's had to have organs removed, silly as it may be.
After waking up shocked to discover that his appendix has been removed, he tearfully takes it home in a jar and buries it in his back yard. Fortunately, Angel!Pinky, along with all his baby teeth and tonsils, visit just before ascending to heaven and gives Rocko a proper goodbye.
At the end of the first half of Crusin', Rocko, fuming about how bitter and misanthropic Heffer's grandfather is, gets some advice from another elderly passenger that reduces him to tears.
"You're going to be old one day, junior, if you're lucky. And then you too will feel the heartbreak of a prune diet or lost memories... the pain and loneliness of losing a loved one. Maybe then you'll understand why old-timers aren't always the bright ray of sunshine you find yourself to be, eh boy?"
In "Kiss Me I'm Foreign", Filburt has to pretend he's married to Rocko to prevent him from being deported. A few scenes later, he has to break up with Dr. Hutchison. Seeing the normally perky Hutchison and Filburt cross paths with sadness in their eyes can get rather depressing. Thankfully, they get back together in the end when Rocko gets his green card.
The final scene in "Leap Frogs". After an entire gut-bustingly hilarious episode of Mrs. Bighead trying to seduce Rocko, she suddenly becomes a sympathetic figure when Mr. Bighead comes home and insults her openly. Rocko then, despite having a great disgust for Mrs. Bighead, kisses her on the hand and tells Mr. Bighead that Mrs. Bighead is a fine woman who can have any man she wants. Mr. Bighead gets jealous, but then realizes that Rocko is right and Bev and Ed make love by breaking plates with their tongues). The scene is a defining moment for Rocko, but one that many for years didn't get to see. It was a banned episode (only after three airings) but probably one of the finest of the series.
The scene in "Power Trip" where Really Really Big Man has Rocko take a look inside his nipples to gaze at his future self. Rocko quickly becomes ashamed of what he has become and tearfully hugs Filburt when the latter walks in all bandaged up as an apology for the way Rocko treated the turtle.
As often as Ed Bighead is depicted as the grouchy Jerk Ass he is, there are a number of episodes showing a more humble side of him and just how he came to be so bitter. He's feeling the stress of age and not looking like the toad he once was, he has a strained relationship with his only son and doesn't seem to have much to live for besides his loving wife. "Nothing To Sneeze At," the episode where Bev gets a nose, makes it clear just how much worse he'd be off without her: the two of them cry when Bev realizes her sense of smell means she can't be around his horrible body odor and he has a literal meltdown (as in he melts into a puddle of goo from shock) when he momentarily thinks that she's died! The old timer's speech in Crusin' could easily apply to Ed: he isn't the way he is without reason.
George: Now, Heffer, I'm still your father. Heffer: You're not my father! You're just some jerk in wolf's clothing!
In "Rocko's Modern Christmas," Ed makes sure no one goes to Rocko's party. Worse, no one cancels on Rocko, leaving him waiting with Spunky for hours for someone to show up. Then he tries to call his family, but all the lines are busy.
Rocko: And it didn't even snow. I don't think I'll ever find any Christmas cheer.