The Absolute Best, in my travelled opinion, are the restrooms in the administration buildings of established universities. The newer ones are ok, but the older ones have lots of toilet choices, big stalls, tall toilets, cold clean tile, and plenty of toilet paper. They are usually in immaculate repair, no annoying graffiti, and the hooks for hanging coats and hats, are always there. I had a favorite, and if I hit it on the weekends, had a window in the stall, so you could look peacefully out on the quad, on a beautiful spring day. Marvelous.
Back to reality, if you have to go while you shop, the department stores are sometimes passible, but the grocery chain toilets, in the back for employees are usually ok. They'll keep them clean before they will the ones up front.
Come to think of it, there have been a few places I wouldn't take a dump. When I was in Turkey, there were some businesses that had a toilet that was just kinda out in the open, just sectioned off by hanging rugs and had no water hooked to it. I think it just drained out the back of the building... I decided I could wait.