This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / The Breakfast Club
Andy's monologue about how he regrets bullying a weaker student and how his dad pushing him so hard to succeed at wrestling has made him not even enjoy the sport anymore. Even Bender can't look anyone in the eye during his speech because it's so sad.
And afterwards, when I was sittin' in Vernon's office, all I could think about was Larry's father. And Larry havin' to go home and... and explain what happened to him. And the humiliation... fucking humiliation he must've felt. It must've been unreal... I mean, I mean, how do you apologize for something like that? There's no way... It's all because of me and my old man. Oh God, I fucking HATE him! He's like this... he's like this mindless machine that I can't even relate to anymore. "Andrew, you've got to be number one! I won't tolerate any losers in this family. Your intensity is for shit! Win! Win! Win!" You son of a bitch. You know, sometimes, I wish my knee would give... and I wouldn't be able to wrestle anymore. And he could forget all about me...
Pretty much the entirety of the climax once the kids have settled down following the scene with the pot. They vary from being at each other's throats to genuinely caring for one another.
After Allison says, "When you grow up, your heart dies." John, stung but trying to act cool, flippantly asks, "Who cares?" Allison's tearful reply, "I care◊", gets me every time. It shows that the compulsive liar of the group is opening up her heart in the sweetest of ways.
In the same scene, Claire's tearful declaration that she will never be like her parents.
"Oh god, are we going to be like our parents?"
"Not me, never."
"It's unavoidable. When you grow up, your heart dies."
The part when you realize that Allison is absolutely telling the truth when she claims her parents ignore her. Watch the beginning of the film again, and see the scene where her parents just dump her and drive off without so much as a word.
Andy: What do they do to you?
Allison: They ignore me.
Brian's story, when he breaks down after receiving a failing grade and it is heavily implied that he wanted to kill himself. He also freaks out about the elephant lamp.
The scene where Bender is acting out his father's mannerisms. The amount of detail and the way his voice gets louder with every word and movement is heartbreaking.
Also, any scene where Bender's home life is brought up. When Andy tells him he's lying about his father's abusive behavior, Bender freaks out and goes off to sit by himself. When Vernon threatens him, he looks like he's about to cry. To anyone who has suffered from abuse at the hands of a parent these moments are definite tearjerkers.
When Vernon shoves Bender in the closet and actually physically intimidates him, it's one of the few times we see Bender being genuinely vulnerable. His small, scared, "You're threatening me?" takes the cake.
Pay attention to Bender's body language during that scene: for the first time in the movie, he's cowering like a frightened animal. That's the body language of someone who's been physically abused. The fact that Vernon not only ignores it, but keeps going, is nightmare fuel in and of itself.
"Look at him. He's a bum. You wanna see something funny? You go visit John Bender in five years and you'll see how goddamn funny he is. What's the matter. John? Gonna cry?"
When Bender yells at Claire about her comparing herself to him, as well as her being prone to giving in to peer pressure.