When you're dumping out your knapsack at the end of your camping trips, you probably leave your sleeping mat in the stuff sack and stash it with the rest of your camping gear. Maybe you store it in the basement, the garage, or if you are like us... you unroll it, let it dry out, then pack it back up and store high up in the closet. It turns out that's not the best way...
Sleeping mats range from water resistant to water proof, so airing them out is a great idea, but only part of the solution. Wetness from the water that sloshed on it from the bottom of your canoe will eventually evaporate, but there is moisture inside from the puffs of air you blew into it to help inflate it.
If you have a self-inflating mat, the best way to store it is to make some room for it under your bed. There isn't much of a chance it will sustain heat damage under there, and you can lay it out nice and flat. Be sure to leave the valve open as this will give it a chance to 'breathe' and prevent mildew from forming inside. This also allows it to spring back into shape better when you take it out for your next trip.
If you have a closed-cell pad, you can leave it loosely rolled (or folded) up, just don't keep it cinched up with a strap as it can damage it long term.
Our Therm-a-Rest self inflating sleeping mat has lasted forever, in fact it was a hand-me-down from my parents and storing it properly has led to years of trouble-free nights of great sleep.