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Tear Jerker: Ruby Gloom
If any moment in the series deserves a spot on this list, it's this, even though you won't really get it until you watch more than the first episode. In Gloomer Rumor, even though it can probably be chalked up to Early-Installment Weirdness, that Misery, the character who can shrug off things that would break most people emotionally and most likely physically, openly weeps at the thought of Ruby moving out. The one main character in the entire series who is used to horrible things happening to her and is able to get passed them without any trouble, practically breaks down when one of her friends might be moving out.
Most of "Hair-Less" is also a Tear Jerker. Skull Boy moving out is the one thing that can make Ruby, the quintessential Perky Goth, forlorn. And not only is Misery also distraught by this, she actually tries to cheer up Ruby.