Store your pocket squares either folded in a drawer, away from light and dust, or hanging using pocket square clips.
Ironing pocket squares or scarves after use is acceptable and can actually prolong the life and shape of the square. Be sure to use the silk function on your iron. DO NOT iron the rolled edges, as this will cause them to lose their rigidity.
Generally, pocket squares do not need to be cleaned. They are decorative pieces only, and should not be used to blow noses or wipe sweat. If however, a piece should need cleaning, dry clean only.