Rapunzel does cry at really tragic stuff, but she's not really that easy to move to tears. In either direction, happy or sad. She doesn't cry when she sees the lights, or when Gothel tells her she's never leaving the tower. (Although that is partially, if you ask me, a mix of having a plan and being in shock.) She does cry when Eugene dies and a bit again when he comes back. Also she was totally choked up in Tangled Ever After and it was adorable. But yes. She really only cries for the big stuff. (Including when she cries on her first time out of the tower; I would count that and thinking about the effect it could have on one of the only two relationships in her life as big.)
Ally has not cried in canon, but I definitely think she would cry on big stuff and probably medium teenage-drama-y stuff. I... Am not sure what I think about her and tears of joy, but she is definitely not a frustrated crier because she spends so much of canon frustrated and... Not crying.
Chel I legitimately cannot imagine crying, and I see a lot more in her than most people do. I don't think she'd let herself cry, really, because Chel doesn't like appearing vulnerable.
Hiccup cries, although I don't think he does in canon even when things get horribly bad? I just can't see him as a character who wouldn't cry on bigger things. Like, I think after he figured out all of the aftermath of the big battle, and it had really sunken in that he lost his leg and almost died? He probably teared up. He doesn't cry when frustrated (he snarks). He also doesn't cry when happy. Bonus: I do not think he cries about his mother; based on the helmet scene, I don't think he remembers her much at all.
Shrek (musical-verse) is weird to think about, but he might be more likely to cry than the others, really. I know in movie-verse he cries in the... Fourth? Film when he thinks he lost his family. Which is admittedly a HUGE THING, but anyway. Musical-verse Shrek is more emotionally open than movie-verse Shrek, so I can see him crying for sad things and possibly really happy things. (Like if he had the kids in musical-verse? There would definitely be proud papa tearing up moments.) He doesn't cry when frustrated, he. Lashes out instead.