Isaac - in addition open up the plate at the back of the guitar and make sure that the hole in the back of the tremelo block is clear of burrs and the string can fit through cleanly,i had a "fulcrum" bridge guitar which would trap the string before the ball seated - you'd tune up and the string would come loose as it pulled up slowly further up the hole -this only happened on the 6th string,they hadn't made the hole wide enough.

David, it's basic but check your nut isn't obstructing your stings/there isn't any dirt in there and that your bridge/pins are clean/seated correctly.