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TLS addict has the best posts of anyone on this board and the above post by Bill D is almost angry given his normally soft sided nature. Bill, I like your more honest realistic side.
I do the underside once a year as part of the winter clean-up. I raise the car up on jack stands at the hockey puck locations and take the wheels off. The wheel wells get scrubbed with a brush and I go over the aluminum suspension parts with scotch bright and soap. The trans/diff etc. get sprayed with a strong soap soultion and the rest of the bottom gets a going over with a soft brush and soapy water. The soap is just Dawn dish washing soap. Simple Green is too harsh on the aluminum parts. My sprayer is ancient but you can probably still find something similar at Harbor Freight. It connects to the air compressor and has a siphon tube that you put into a container with whatever liquid you want to us as a cleaner.
The whole under side gets a good rinse afterwards and I scrub the insides of the wheel barrels before the wheels go back on the car. It's a messy job and my wife thinks I'm nutz... she's right but it has nothing to do with detailing the underside of the car.
I do this annually as well. I use a cleaner/degreaser instead of dish soap but essentially the same process. For me it's not just a Corvette thing. I also do this to my daily driver. I guess I'm just a neat freak... and a little bored.