7. The trem post anchor [insert] has become loose. This is the most common problem I will run into on older guitars, primarily basswood of course. Most types of tuning problems will give predictable trem return. With loose inserts the return is erratic. The trem posts screw into a metal anchor that is press fit into the body wood. Over time this anchor can become loose, especially on guitars where the trem post set screws have not been tightened allowing play between the post and anchor, and especially on basswood which is a softer wood. Remove the trem and with the post set screws tight try to move the posts [in the headstock to tail direction and back] with the 4mm Allen in the top of the stud to see if there is any play. Keep your eyes on the anchor checking for any play. If they don't move use a junk stud, or put tape around the top of the stud, and try again using pliers. [I've had many you couldn't move with the Allen alone but pliers showed how loose they really were. If you don't have a junk stud take the stud out and use anything [4mm Allen wrench from the stud, a bolt with the right thread, etc] put it in the hole and push the anchor toward the head, and back toward the tail. Even a little play will cause tuning instability, the anchor must by stable. There is over 100 pounds of string tension that makes the anchors move, test using the same kind of force] If they are loose you will need to pull them out of the body. Make sure to cut the paint away from around the top of the anchor 1mm or you will tear the paint up with the anchor as it comes out. Grab the post (preferably using an old junk post to use just for this, or a bolt with the correct thread) with a pair of pliers and work it out of the body. [I use a cats paw [tool for pulling nails] with shims laid across the body to protect the finish] If the play is minor give the headstock side of the hole a superglue coating, then use accelerator to cure it instantly. Check the fit with the insert, if it's still loose give it another coat of superglue, repeat. If it's tight use a good quality wood glue dabbled around the inside lip of the hole press the anchor back into the body the same way you pulled it with pliers on the post. Let sit for at least 24 hours before reassembly. NOTE - do not make the fit too tight or you may crack the wood behind the anchor [between anchor and trem rout]. Basswood has very little resin and after 10-20 years if plenty dry and prone to cracking if you give it a reason. [Many lower line guitars do not have post anchors. Some screw right into a metal plate that is screwed to the body. You get what you pay for.]
You'll have to do the same thing if your anchor is tight, but spinning in it's hole. Sometimes a set screw can be so tight you'll bend an allen wrench trying to get it loose, and turning the post without loosening the set screw causes the anchor to just spin. The only way to free the post is to pull the anchor out so you can grip the anchor with pliers while wrenching out the post. Reinstall as above.
If a loose anchor is neglected long enough it can actually oblong the hole, sometimes to a great extent. You can actually see that the post is not at 90* to the body. If the oblong is under 1mm using the superglue and wood wood glue will fill it enough. Be careful not to use too much glue or when you press the post back in it will be forced in through the hole in the anchor's base. Use plenty, but not too much
If the oblong is larger than 1mm you'll need to use a good non shrinking acetone based wood patch on the neck side of the hole. Pack it in, most at the top than in the middle of the hole, but not too much. While it's still wet [work fast, wood patch has a tendency to dry very fast] press the anchor back into the hole and seat it. Try to press it in at a perfect 90* angle to the body. When it's seated pull it straight back out keeping it at the same 90* angle. Let the patch cure for 24 hours and then reinstall the anchor using wood glue as described in the first paragraph. [Always check the bottom of the hole for excess wood patch that has been forced down there when "forming" the new hole. If there's too much it will prevent the anchor from fully seating. Use anything to scrape it out so the anchor can fully seat]