This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / A Certain Scientific Railgun
Episode 5: Watching a little Uiharu crying for help as she watches her friend Kuroko getting beat up by the bank robber, you can't help but to cry yourself and wanting to hug Uiharu to calm her.
Uiharu telling Saten not give up as she will wake her and their friends before they succumb to the side effects of Level Upper. Saten is grateful of what she said before she falls into a coma.
At the climax of the battle with the AIM Burst Misaka starts to hear the collective thoughts of all the level zeroes who wanted to become espers. As epic as the battle is, listening to the monster's laments and fears of never amounting to anything or achieving their dreams juxtaposed with with Misaka's gentle words of understanding and encouragement to keep on trying makes for an oddly touching moment.
Uiharu and Saten's reunion after being cured from Level Upper and Saten realizing there are some things more important than powers; her best friend. Awww...
Episode 18: The fearsome Tokiwadai Dorm Mother is dressed up in her finest, ready to confess her love to her co-worker Mr Daigo at the Kindergarten Orphanage; Only to have him innocently ask as a friend advice on proposing to their significantly-older boss. For one terrible moment you can see the exact moment her mighty-heart breaks, but with kind dignity she congratulates her love with a warm smile instead.
Misaka 9982's final moments before her death is just, so, heartbreaking. She's lost a leg, and is crawling towards the Gekota pin Mikoto gave her. She picks it up and smiles...and then Accelerator drops a train on her.
Oh, and Mikoto saw this entire thing. The next chapter starts with a brief flashback to her speaking with Kuroko about the possibility of clones.
Kuroko: What would you do if a clone of yours appeared before you? Mikoto: Ah, that would freak me out. I think I would wish it would just disappear. [present: Mikoto goes ballistic on Accelerator]
Seeing Accelerator's flashbacks when he first kills a sister and realizes that he needs to kill her (rather than simply overpower her). Later before being knocked out by Touma he snaps out of it asking himself "What am I doing?"
What happened to Kiyama's students and what she had to suffered to saved them makes you want to support her.
But it gets better later, after several setbacks, Kiyama is able cure her students. The moment she shed her tears, you can't help but cry too.
At one point, Mikoto screams at 10031 "Stop appearing before me with that face, that voice," since it reminds her of all the other clones she's failed to save. Heart-wrenching enough on its own, but 10031 interprets this as Mikoto hating her, an assumption that is seemingly confirmed when she asks the researchers how they would react to their own clones.
Misaka 10031: To Onee-sama, Misaka is an existence she wants nothing to do with.
Railgun S Episode 16: Watching 10032 go to the vending machine and get the same Gekota pin which was given to 9982 just reminds you that because all the clones share memories, means that each one of them carries with them the pain of the 10031 dead clones before them. But even a single keepsake shows you that the dead clones live on via the MISAKA Network.
Chapter 49: Kuroko, Uiharu, and Saten all forget about their friendship with Mikoto, leaving her alone all alone, because of The Queen's power. Plus her Sister being kidnapped.
Chapters 59 and 60 showed us what Misaki was like when she was younger. She made an actual friend named "Dolly", but was absolutely devastated upon seeing her collapse and later learning of her death. What makes it worse is that the scientists were intending to dispose of her if she wasn't living up to their standards. A broken cutie indeed.