A good number of the P&S models have optical image stabilization. Only the cheapest models are missing that.
Speaking of auto stabilization and auto focus, my experience is that many people forget (and it also happens to me when in a hurry) that for many smartphones it takes a few tents of a second to get that stabilization and auto focus going.
The auto features that are there work really well when you are paying attention to that delay, but on many situations, the camera is moved away from the subject too soon by the operator thinking that the picture was already taken, the result is there for even the OP to see.