A bit late to this discussion as I've only just became a member. Maybe my input might hopefully help someone else in the future?
My 2001 C5 Targa Auto does the same if I don't drive the car for a long period of time.
Same as Brads. When I start it, I get a small amount of fluid on the floor. Not all the time though! It only does it if I put my car in Drive or Reverse within a minute or 2 of starting it up (because I'm usually just moving the car around on my driveway). It will leave a small little pool of oil around 50ml or so.
I recently had it up on a 4 post ramp to trace this issue and found the gearbox, diff and driveshafts are all dry.
I did have a lot of oil covering the rear leaf spring and managed to trace it right up to the gearbox breather pipe located above the spring.
My theory (being a motorcycle mechanic and quite handy with cars/vans also) is the following:
The oil level for the gearbox should be checked on a 2 or 4 post ramp, with the vehicle flat and the engine running.
The fill hole is on the side of the gearbox sump and the oil capacity level is basically fill the gearbox through that hole, until oil comes out of it and you can't get any more in.
You then wait until the gearbox oil stops dribbling out, select the gearbox through its gear selections a few times (car still running), back in to Park, double check the level and top it up if required... it is then full. Fill plug back in. Winner

Now when the car has sat for a while, you have to remember that the gearbox oil cooler is located right at the front of the car and it sits slightly higher than the height of the gearbox itself. So when you leave the car for a long period of time, the oil that is pumped from the gearbox, all the way up to the front of the car, through the oil cooler and all the way back will in theory drain back to the gearbox itself (gravity fed basically).
This then makes the oil in the gearbox OVER its capacity! BUT DON'T WORRY!!!!! I have kept an eye on mine for a while now and found that when you start your car again, the gearbox will slowly pump the oil back through to the cooler as it should and then your level will be correct.
As mentioned before, if you start your car after leaving it a long time and then put it pretty much straight in to gear without giving it time to circulate the gearbox oil back up to the cooler, then it will assume it is overfilled and chuck out the excess through the breather... obviously this doesn't happen when the car is being used every day or at least in occasional use (I have noticed) as it takes a long time for that oil to drain back from the cooler, to the gearbox when it isn't in regular use.
All I can say on the matter is (and feel free to correct me if I'm wrong, but this is just MY personal experience and MY findings, on MY car) if you've left your C5 sat for anything over let's say, 2 weeks? then give the car around 10 minutes or so after starting it to just sort itself out before driving it or even putting it in a gear. You should find that no oil will be on the floor

Temperatures and Pressures as they say.
If you're concerned about the oil level being too low already and don't have the means to get under the car yourself then drive it to your local trusted garage where they can put it on a 2 or 4 post ramp (because the car needs to be perfectly level remember), ask them to leave the car running in Park, take the fill plug off the gearbox and top it up if necessary. You won't need to run it through the gear cycles as you're only topping it up... but it won't be anywhere near as much as you think.
Sorry for my LOOOOONG and probably boring first post, but I hope it's informative and clear enough to anyone suffering the same issue.
This is only what I have found in regards to my own C5 Vette

