Just got a door sling and before we even got started I fell out of it as I was adjusting myself. Luckily I'm a gymnast and just grabbed onto the straps to slow my fall. No injuries and we enjoyed the swing afterwards.
I actually pulled my shoulder out on ours. I guess I was holding it in a strange position or something. I think I was trying to reshift my body and pulled up on the swing and pop went my shoulder. I was in a sling for about a week. I came up with some crazy story for the doctor. I'm super clumsy though, so I think for most these would be safe.