This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / 21 Jump Street
Blindsided: A teenage girl who's being sexually abused by her father, repeatedly tries to get help from police and social workers for her home situation, but they repeatedly fail to help her - because her own, abusing, father is a police Captain and uses his police influence to habitually dismiss her case. It's made even worse by the fact that the girl also has lost her mother already, that the father has recently started abusing the girl's younger sister too, and that at the end he's arrested for it (which might be a good thing in itself, but means the girls now don't have any parents left).
Blinded by the Thousand Points of Light, which was about homeless teenagers. Seeing how hard life is for them and how they struggle to survive is very sad indeed.
A Big Disease With a Little Name, which featured a teenager with Aids (back then a death sentence, and an even bigger stigma) who was shunned by his peers and coming to terms with his dying. The issue may seem obvious and dated this many years later but if you can get through "Just hold me, man..."
Loc'd Out - Part 2 and Draw the Line - Hanson is wrongly convicted for a murder he didn't commit, and then is shown in prison, where of course he has an extra hard time because he's a cop...
Judy Hoffs' love life in general, somehow, is sad—even though she's presented as a very attractive lady and is a Ms. Fanservice and has lots of male romantic admirers. She is about the only protagonist who never gets a (happy) relationship in the series note Hanson has at least two long-time girlfriends during the series, plus some flings; Doug had Dorothy (though not happy overall, at least it was steady) and later the love of his life, Marta; Harry is shown to be on some happy dates, and Fuller too. Amongst others:
What About Love?: She gets into a relationship with a (sort-of) co-worker, but then it turns out he kept from her that he is actually married; after which he turns into a Stalker with a Crush, which is a Nightmare Fuel experience for her.
Stand By Your Man: She is raped by a suspect she needs to interact with while undercover.
Research And Destroy: Another time, a man she interacts with while undercover, stalks her; this, in contrast to the aforementioned case, is Played For Laughs as opposed to Played For Drama, and doesn't lead to rape, but it still greatly annoys her.
Chapel Of Love: The date she goes on on Valentine's Day turns out to be extremely boring and a big disappointment and she cuts it short to go back to her work/co-workers.
Fun With Animals: She might or might not have gotten together with her co-worker Dennis Booker, who might or might not be actually racist (all of this is left ambiguous). Anyway, she only gets flack from her other co-workers for (possibly having done) this, and it doesn't lead anywhere.
She seems to have some sexual/romantic tension with co-worker Hanson in the first half of Season 1, but this is completely abondoned later on. Then in Season 2, out of nowhere, she almost sleeps with her other co-worker, Doug, something she afterwards feels extremely awkward about and wants to forget forever.
Tom Hanson did have (mostly) happy relationships with three different girlfriends over the course of the series. However, the way each of those romances ended was very sad:
Amy gets murdered in front of him, after which he develops PTSD. Worse, he was actually going to break up with Amy that night and he feels terrible guilt not only about the fact that he, as a cop, hadn't been able to prevent her murder, but also about that if he'd broken up with her earlier, she wouldn't have been where she was shot dead.
Jackie betrays his trust, which hurts him deeply, and he immediately ends it with her. She begs for forgiveness, but he explicitly answers he can't and won't forgive her; she walks away guilt-stricken.
He almost gets back together with an ex-girlfriend, Linda, whom he runs into again. He makes it clear that he thinks of her as "the one". But she tells Hanson she can't be together with him and leaves town, even refusing to tell him where to or giving him contact info, as she want to cut all ties with her past.
The death of Tom Hanson and Doug Penhall.
Jenko's feelings are really hurt when he overhears Schmidt make fun of him.
"You still don't know your Miranda rights, you idiot!"