Hello, IHATECOLITIS.
I don't carry wipes for the seat but, it's funny, I don't know any woman who doesn't put down a layer of toilet paper on a public toilet seat before sitting down -- even my mother, who is in her 90s! Maybe it's a female thing? The grimiest object in a stall would be the lock, I think. And the flush handle, if it isn't automatic flush. And the floor of the stall. So we use a bit of paper towel or tissue to open the door on the way out. Most public places have caught on and put a trash bin next to the exit. I'm curious how many others put down a layer of paper.