Good question, I've recently put up a rope type swing for my son. It consists of two ropes, one on each end of the bench style seat. Regarding the loop/hitch to the branch, on one rope I took a round turn and made a poacher's loop, on the other rope I tied the constrictor hitch. Below, where the rope runs through the two holes on the bench style seat, I just tied a double bowline on each end. It's held up really good for my three year son but I'm not particularly happy with the system of knots. IMO, the very best solution for the rope to the branch scenario is by tying off with a Bull Hitch and this is what I plan on doing. The double bowline's have worked great though, they are of course backed up with an OH stopper knot.
As for tying off to an anchor, I'm a fisherman and have four anchors in my boat, don't use a loop (bowline) as your knot. An ideal anchor knot is the buntline, lobster bouy hitch, or probably the best of the lot is the anchor bend variant. Two reverse half hitches work good too, it's probably the easiest one to untie if you're worried about untying the anchor at any point. The reason I don't use a bowline or loop knot down there is because it just creates a snag opportunity, hope that makes sense.
I don't find myself tying loops that often but when I do, it's usually the double bowline for the stationary type and the poachers knot for the slide and grip type.