Are heavily worn rims safe to use on trainer?
I don't see why not, unless you plan on braking while on the trainer? Theoretically you could still have a rim failure because the wheel/tire pushes against the resistance wheel. However, this is a constant tension, not sudden impacts. If a spoke were to become loose though, the rim could fail and the wheel could implode, but since your on a trainer it's not really all that dangerous (besides picking bits of wheel out of your drivetrain).