This is based on opinion. Please don't list it on a work's trope example list.
Tearjerker / Shikkoku no Sharnoth
The deaths of all the birthers of Metacreatures. Jane Doe in particular.
Elly leaving England. She's certainly a smart kid, she knows and accepts what happened to Jane, but she's still a child going off on her own to a foreign country and an uncertain future while being pursued by adults who want to harvest her brain.
Moran. Everything to do with Moran:
Her Anguished Declaration of Love, the night before her death. It's the only time we see her crying, putting aside her constant denial for once and admitting to how truly miserable, lonely, and guilt-ridden she is.
Moran: Forgive me...please forgive me. Forgive me for yearning even more shamelessly and greedily than I ever did when I was human.
Moran's death. It's even worse considering how fine she is with her own demise, comforting a very distraught Mary.
Moran: I apologize for startling you, Mary. Thank you, for finding me.
Her last words to M.
Moran: Master, have I been of use to you?
Moran very nearly crossing the Despair Event Horizon several times throughout the story. It gets increasingly worse when not only does she feel jealous of Mary, but then Jane Doe comes into the picture, and she's also clearly hated by other members of the Society.
Moran gets so desperate for M to acknowledge her at times, that she starts asking to be rejected, because even negative attention is at least some kind of attention.
Moran: Please blame me for these unauthorized actions. Please blame me for the abnormality in my circuits. Please blame me. Please reject me.
Her backstory. She is the only one to survive a Metacreature attack on her whole town. When M finds her, she's little more than a broken, starved, exhausted little girl who has lost all hope and faith.
She does absolutely nothing in her spare time aside from polishing silverware and inspecting cigars, sitting by herself in a hotel room and waiting for M. Never resting, never amusing herself, just spending night after night all alone, sometimes for days or weeks at a time.