In sport and trad climbing, a sit harness is widely used (except for children). However, I wondered if there is the possibility to fall out of it?
One situation where I could see this happen is in climbing a roof, when the hands lose grip on the holds first and you fall head first.
No you cannot - as long as you fit it correctly.
One test you have to do when adjusting the width of the hip strap, is to pull down on the harness to make sure it is impossible to pull it over your hip. So no matter in what weird position you fall, the harness can neither move up (leg loops) nor down (width of hip loop as explained) significantly.
Additional infos why you might not only want to use a sit harness:
The main reason of using an additional chest harness or combined harness is to keep an upright position without any work. In a sit harness you need to grab something (the rope) with your hand or use significant body strength to keep an upright posture. A correctly fitted chest/combined harness does this automatically.
It is also a good thing to be upright during an uncontrolled fall. During the free fall phase, a combined harness wont help for that. However as soon as the rope starts braking you, the pull will get you into a (close to) upright position with a combined harness. This is not important because you might slip out of the harness, but because it is easier to "end" a fall gracefully (i.e. without injury) when upright and feet first. Falling down back or head first obviously increases the chances for serious injury during impact. When I am talking about "end" of fall/impact I mean protruding rock features and/or the moment when you hit the wall horizontally (more or less) once the rope stopped the vertical movement (which almost always happens, unless you are in a big overhang).
You certainly can, if the waist belt isn't high enough and tight enough.
A few years ago I briefly worked at a small rock wall where among other duties I belayed and showed people how to put on a harness. In my experience most newcomers don't (want to) fit the waist belt properly, and a significant fraction of regular climbers fail here too.
The waist belt needs to sit above the hips, not around the hips, and certainly not falling off the butt as some find fashionable with pants. It is not sufficient to casually tug at your harness and say "look it doesn't slide down." In a fall forces many times body-weight can occur: the padding will compress, the textiles will stretch, and your body will deform. The harness needs to hold on the pelvis not on the buttocks or other fleshy bits.
Look at images of the human skeleton: the thick part of the pelvis is at the top, the iliac crest. That is where the belt should hold. If it slips (or starts) below that point only the projections of the femurs exist to catch it, and they are mobile and of insufficient dimension for the task.
