Angelica and P-chan, her new bird, in the very first chapter. You can see the ending coming a mile a way, where in trying to secure a mate for his company, Angelica ends up accidentally killing him since the bird species involve Black Widow mating. Note: this does not help!
Mellow: He... promised... If there's no... Jody... There's... no... me...
Chris' adoptive parents arriving at Christmas and demanding to be allowed to take him home, bringing up all the logical reasons he shouldn't be living at the shop. It just kills you, because you know they're right, and yet you also know they're dead wrong.
The entirety of Catherine's story.
The finale; D's father's 'last memory', Q-chan's true identity, the contents of D's suitcase, and worst of all, grown-up Chris meeting the new D.
Chris:I was wondering... 20 years ago, my older brother, Leon, left to return something to Count D. Something that the Count had left behind. Do you know if he ever made it?
The entirety of the Devil chapter...how you see the grandmother regress to her old days and finding out that her cute precious teddy bear was a pet her father signed a contract with, and it kept a bible and a photo of her past inside it during the beginning of World War 2, so that she would survive the Holocaust. Oh god, it makes me cry right now.
This troper will raise you the spin-off story from that, dealing with Count D living in Germany while World War II is going on. Especially the scene where D and Alex stand on a statue and talk about how the humans might kill each other all off after all, leaving them to face away.
That. Fucking. Butterfly. Just his FACE first when he realizes that Chris doesn't exist, and then when he can't find the shop anymore.
For that matter, the SECOND time Leon goes to the shop and finds nothing there, and this time for real as Count D has packed up and left. His absolute refusal to accept what has happened is heart-shattering.
D and his "sister". Papa D is an asshole.
Monica. And the fact that apparently D's father, despite having no great love for humans, was saddened by her passing as well.
The story of the child who was trained from infancy to be an emotionless assassin. He was raised believing that his own mother gave him to a criminal ring that bought children for that purpose. When he hides in D's pet shop and tries to escape out the back, he finds himself in a room that has a woman in it. D says that the woman is the boy's mother, and the boy is so angry at her for what she did that he tries to shoot her. D stops him and reveals that after she had him, his mother was too ill to care for him. Until she recovered, she left him in the care of friends. It was those friends who betrayed her trust and sold him, leaving his mother to slip into insanity over the loss of her child. The boy is so shaken by this that he tries to kill himself. The story fortunately ends on a heartwarming note, as the boy survives and D removes the black scorpion tattoo (the only mark identifying him as a wanted assassin, instead of a regular punk who was just guilty of attempted murder), meaning he'd have another shot at a more normal life.
The story of Robin and Medusa. Especially their deaths.