So far, the ones I've met are:
University Students
- The BFF (prologue)
- The Roommate (prologue)
- The Roommate's Partner (met randomly one morning)
- The Chipper Student Advisor (the tour guide from the bus tour. So far only encountered randomly)
- The Classroom Admirer (met by attending class)
- The Classroom Harasser (met by attending class)
- Dorm RA (unconfirmed)
- The Rich Kid (unconfirmed)
I've seen stuff about the Dorm RA in the comments, but never met them myself.
Sports Teams (Work In Progress)
- E-Girl/E-Guy (E-Sports team, gender variance unconfirmed)
- 360 No-Scoper (E-Sports team, team captain)
- E-Sports Coach (unconfirmed)
University Staff
- The Summit Market Manager
- The Mail Centre Worker
- The Campus Clinic Doctor
- The Campus Clinic Nurse
Riverside
- The QuickieBurger Franchise Owner
- The QuickieBurger Organiser (student at the university, don't think I've seen them around though)
- The QuickieBurger Coworker (same as above, not sure if I got their title right)
- The Tattoo Artist (met only by going to the tattoo shop, afaik)
- The Piercing Specialist (as above)
- The Parasocial Fan (met via a random town event after getting streaming popularity or at the River Rat)
Note: these are all of the special (customisable) characters that I have encountered currently as of my previous and current runs. I'm not that far into my 0.4 savefile yet, so there may be more that were added and I may not have even met all of the ones in 0.3. There's also not content available for all of them, so far as I can tell.