Spirit offers perhaps the worst airline experience in America. But Spirit will let you pay to bring a full size carry-on bag onboard.
United’s Basic Economy fares won’t let you bring a rollaboard onto the plane (unless you have MileagePlus elite status, a United co-brand credit card, or are a Star Alliance Gold member). You have to check the bag (and wait for it, and maybe lose it) and pay for the privilege of doing so.