On a day-to-day basis I carry a cross body bag with an adjustable strap that can quickly become a fanny pack of sorts. I do this at our home amusement park for rides where I am restricted from caring a bag.
I was hoping to work in new bags for the Disney bounding my daughter wants us to do. I know on rides like Peter Pan the bags can just sit on the floor. What about things like Expedition Everest and BTMRR? It is possible but my mother would stay off the ride and hang out with the bags, but at the same time she enjoys shopping at that point in having a bunch of garbage with her would be annoying.