Uh yippee... the chains vs straps debate..
and now you've added the axles vs frame argument :beatdeadhorse: 
From all the threads I've read on this... the answer I have come to is.....
just what Matt said. For *our* applications (cars and or trailrigs) either proper straps or proper chains and binders are more than sufficient to safely secure and hold in most but the most extreme situations. Same with to the axles or the frame. It comes down to what you feel the most comfortable with.
*I* like my straps on all 4-corners, crossed in the front (because I always have) to the axles. It has always been secure enough for me and I've never had and issue unless I cut corners.
I am considering switching to chains with axle hooks, or pre-set straps with axle hooks in the back just to save hook up time.
Many folks strap to the frame with no problems, I don't like how the bounce in the suspension loads and unloads your binding method, and if you don't have positive locking attachments, there is a chance of it coming unhooked. (even if you since the truck down to the bump stops, there is still a degree of bounce, and it can't be good for the suspension). *I* feel the axles are a much more secure place, but if you want to strap to the frame, go for it if your comfortable.
________
CHEAP VAPIR VAPORIZERS