I've had my spare tire on the roof for about 4 months now.
It sits ON the roof, between two cross bars.
It's held down with two ratchet straps criss crossed over the top, hooked into the cross bar end caps.
In response to AZZY's comments:
3: Anytime you handle a spare tire, you're going to get dirty
2: I haven't noticed and handling (or MPG) loss with it up there (but then I don't drive like I stole it either!).
1: Er, well, the inspection guy may be right about this. I wonder if tires mounted to aftermarket roof racks are any more secure