I actually have both situations for one car.
I have a 315/60x15 drag radials, recommended for a 8.5-11 rim on a twelve. I have seen them on a fourteen. They work best on a wider rim (11.5 tread width)
The street tires I have for the same car, are too wide for the rim. They have 15+inches of tread, on a twelve inch rim. They are run fairly soft, under 20 lbs, to avoid wearing out the center. They will still have plenty of excess load rating, it's a light car.