Not sure what you mean by flat but they are made to fold up in a certain place all the time so not an issue with creasing that I have found.

Flat means a plain Roman, no ribs, no hobbles.

Wrinkling will depend on the fabric. Normally a drapery cotton fabric won't wrinkle but a linen fabric will.

We have flat roman shades throughout our house and haven't had any issue with wrinkling. We open and close them daily, and multiple times a day in the kids' rooms for naps.

This is so helpful. The shades that will go up and down the most are 100 percent cotton. I also have a linen rayon blend for a guest room (will be used 2-3 times a year at most) and a 30% 70% polyester blend for hallway shades. We live in an urban area so I’m not sure how often we would lower the hallway shades. We might just always leave them halfway down as our neighbors look down into the hall.