so i've been ruminating on this question awhile I'M SORRY it's been a few days, and i'm not good with answers unfortunately
but i do like side stories with continuity--where a reoccuring interaction between characters happens. it really does feel like a parallel plot to the main thing then, which i'm a fan of
in particular, i want to call attention to a few side stories primarily centered around rui and meiko/virtual singers.
the very first one of these i would say is rui's cheer squad side story, where meiko/luka with rui overhear his classmates talking positively about him and his work at the cheer festival, and meiko decides to call out to them to get their attention so that rui and them could talk and bond.
the second one of these is amidst a dream rui. in here, rui goes to the wonderland sekai to take a break where he meets with her and luka. he tells them that his feelings have been distracting him and he wishes someone else could control his emotions for him. they encourage him, it's sweet.
the real piece of pie though i think is ohe. in ohe meiko, she talks with him and congratulates him for how he didn't give up and that honest feelings move the heart. it's a bit silly, but i especially love that line. it's good to be yourself and to have some degree of selfishness.

similarly for rui side stories i want to call attention to rui's bfes. side story part 2 has some of my favorite dialogue in the entire game that's made me cry so bad and i encourage you read it for yourself
https://youtu.be/D7c_OW9YxGY?si=W7kub8_pvFfuMCfj

other side stories that make me weepy are pretty much any of the ones with continuity regarding mizuki and rui. hermit's side story is a fantastic companion piece to a wonderful event that i think about from time to time.

unfortunately i've just ended up talking a lot about rui. that's my bad. umm. i like the an/akito+ken side stories that litter earlygame. i think its lyrics (both) -> nocturne (an) -> sbd (akito) -> bout (both). i wish kensan was my dad