Please don't list this on a work's page as a trope. Examples can go on the work's YMMV tab.
The first half of episode 1 easily counts as this, due to it being a very concentrated piece of Break the Cutie. So much that a troper stated "I bet Gen Urobuchi wishes he could crush his characters this completely."
To elaborate, two moments really stand out. The first is when Kotoura's mother leaves. Haruka is crying and begging for her not to go away, insisting she'll be better daughter. Her mother coldly pushes her to the ground, and says she should never have given birth to Haruka. If you didn't tear up even a little when Haruka is left a broken mess, apologizing, then your a cold bastard. The second was the stray cat. Losing it all but broke Kotoura.
While inciting more anger and horror then anything, the fact that Kumiko breaks before the end of the first 10 minutes, is pretty sad. Especially how much she originally doted on Haruka.
Episode 12 builds on this scene, as Haruka sees her mother's thoughts just moments after she left, crying her eyes out at not being a good enough mother for her. While it doesn't justify her abandonment, at least Haruka realized how much her mother regretted her action. Her mother's eyes briefly revert back to normal when she hears Haruka sleep talking that she wants eggs over easy— Haruka's telling her mom about wanting eggs over easy for breakfast many years ago was the beginning of the realization of her telepathy.
The ending of episode 3. Manabe has gotten out of the hospital after a nasty fight with the Moritani dojo's four goons, and Kotoura is nowhere to be found. Even though he's the one still cut and bandaged, his thoughts are "I failed to protect her." Push comes to shove, the wounds he does to himself because he loves Kotoura that much are the ones that hurt him the most.
The first few lines of The Flower of Hope, being that song's own Downer Beginning, is naturally tear jerking. The cynicism the third line is particularly so.