Gaster has extreme claustrophobia, and will flip his shit and start hyperventilating if he's stuck in a tight place.

Papyrus HATES the dark, which is one reason why he keeps the lab in the back of the house so well-lit.

Chara hates bugs, especially spiders.

Toriel is terrified of snakes and will hide behind Asgore if she sees even a garter snake.

Asgore is afraid of losing his family again, and always calls them every two hours or so if they leave the house alone.

Sans is afraid of things appearing out of nowhere, which is ironic.

Undyne is afraid of Alphys feeling bad about herself.

Alphys is afraid of Undyne leaving her.

Vegas is afraid of people being scared of him.

Frisk is afraid of someone else gaining control over the timeline and fucking with it.

Flowey is afraid of Papyrus, Chara, Asriel or Frisk getting hurt.

Asriel is afraid of being turned back into Flowey.

Claire is afraid of her friends falling down the mountain and dying...