We have been storing our wine "right side up" -- which is to say upright, with the corks pointing to the sky. We don't have a storage facility yet for laying them on their side.
Below is advice from the president of the San Diego Amateur Winemakers' Association, who just sent me this note:
"Standing up is only good if up means upside down. The corks need to be in contact with the moisture. Side is best but not really feasible unless you have a chateau and a cave. Wineries age full bottles upside down. After bottling (and corking). Keep them right side up for a week or so to let excess gas get out (and not have push out). Then flip them. Just like anything you might buy on a trip. Travel with it right side up. But after you get home, flip those babies. "