It depends on the bag and whether I'll be using it alot in the next days/weeks. Like the onatah, it rains a LOT in Iceland so it doesn't get used as much as it should.
I keep the original tissue paper that comes with the bags. When I am done using a handbag, I put the tissue paper back in and then put the bag in its dustbag.
My bags are on display on a shelf going around the room above the doors. I stuff them with clean rags and those annoying event tshirts I cannot bring myself to throw out. I do stuff small bags then place them in larger ones. Leather bags can get out of shape if unstuffed.
If I put accessories in the bags I forget where they are, don't do that anymore. Had an epi change purse missing for a couple weeks with some ID cards. Not good! Big sigh of relief when I found it hiding in a pocket this week.
For my structured bags like the Mono Ellipse PM and Damier Belem PM, I just put them in their dust bags. For bags like the Paps and Tango, I use old towels. My Speedys all have Pursekets and cardboard so they just wear their dust bags.
I don't store my speedy or my keepall folded flat because over time that can cause permanent creases in the canvas, so those I stuff with a t-shirt (I use my keepall to store other bags and an LV towel I have). The rest of my bags don't need stuffing.
Sometimes I stuff my Speedy 30 before I put it away, sometimes I don't. Depends on how much time I have and how long I think it will be out of use. I haven't had any issues with it creasing.
I don't think I've ever stuffed my Damier Speedy 25, it holds its shape very well so doesn't seem to need stuffing.
I put the plastic air bags that you get in packaging boxes in mine. They usually are just enough to keep the bags in shape and easy to remove. I take one out of the one I want to use and put it in the one I am putting away. Even the blow up ones that come from Eluxury work. They also come in different sizes and I seem to buy alot on line and always have plenty of sizes. I just think it makes the bag store easier too.