All wine with natural corks should be stored on it's side to keep the cork moist and sealing. If you put them vertical, the cork will eventually dry out and you'll loose the seal and the wine will start to expire.
I don't think this is an issue with the synthetic corks though. I'm not sure so I won't say for sure. But, since it is synthetic, it doesn't depend on moisture to stay moist and maintain a seal.