Papyrus and Undyne are those two friends that will never accept compliments until the other person does it first. This ends in Sans, Chara and Alphys having to calm them down before they start shouting compliments at each other in public in a never-ending loop and attract too much attention.

Papyrus has a huge crush on Undyne, but let Alphys have her, similarly to the Chara-Frisk-Asriel situation (the roles of the situation go respectively), and even helps Undyne look nice for when she goes on a date with Alphys by doing things like helping her pick an outfit, doing her hair, and giving her tips. Then he dies a little bit inside whenever Undyne mentions him as a friend. Friendzoned, ouchie...

Chara knows about his crush on the fish lady, but doesn't mind, because she thinks it's cute. They talk a lot about the people that they got friendzoned by - for Chara it's Frisk, and for Papyrus it's Undyne. They hold no ill will against Asriel and Alphys, but always talk about how lucky the aforementioned monsters are to have their partners.

Because I'm obsessed with the "Jenny" song, Papyrus actually does multiple things that Cherry does - stealing little things from Undyne's bag, such as pens; using her shirt as a pillowcase (although he feels like a creep while sleeping on it); and trying to subliminally get her to hold his hand when they're out together, along with other things - pretending to fall asleep on her; purposely tripping and hurting himself in order to get her attention; seeming sad so she comforts and spends time with him; and even looking lonely so Undyne invites him to come along with her and Alphys as a buddy hangout. He falls silent and lets Undyne do the talking when people mistake them as a couple, because he wouldn't be able to correct them himself without dying even more on the inside.

Alphys knows that he likes her, but doesn't say a word. She doesn't object to it, because all she wants is for Undyne to be happy with or without her, but would be sad if Undyne left her for Papyrus. However, she doesn't hold any sort of grudge against the skeleton for loving her girlfriend, and hopes that Chara can make him happy, which she does.