Starting at the tuner, are you winding the strings in a single neat layer around the post?

The string should have a reasonably sharp bend where it exits the tuner hole so that the string begins winding close to the post. The windings should progress downward along the post in a single layer for about three wraps total. There are a few different ways for anchoring the string to a standard tuner. I will try to post a drawing of the one I use, as it's hard to describe.

Many recommend that the strings be "stretched" after bringing them up to pitch. They are not really being stretched, but just fully seating the ends of the strings at the tuner and bridge. To "stretch" them, grap each string at about its midpoint, and pull it away from the fretboard about one inch a few times. I find that when the strings are properly installed, "stretching" is usually not required.

Also, always tune up to pitch, never down. If you end up too sharp when tuning, lower it back down until flat, and bring it up to pitch once again. Friction and gear backlash can prevent the tuning post from following the tuning gear downward completely when tuning down. Then, sometime later, the post slips and takes up the backlash, and the string goes flat.

Finally, turn the tuning key to wind the string onto the post. Don't just wrap it around the post like wrapping cord around a spool. That will put a twist the string.

Fixed bridges, and decked tremolos that only dive, are generally not sources of tuning problems.

Nuts can be a problem if any slots are too tight, causing the string to bind. In that case the string can end up with unequal tension on either side of the nut. Then sometime later the string slips to equalize the tension and the pitch changes. Slot binding at the nut is especially a problem with tremolos. Even if the strings are not binding, graphite or a commercial nut slot lube can also be necessary.

Floating tremolos are notorious for not always returning to pitch. The problem is usually friction somewhere in the movement, especially with the 6 screw mounted types. Tremolos need to be installed carefully to prevent any friction. There are mechanisms that are designed to always return a floating tremolo to the same position, but I haven't tried any of them.

Exactly what type of tuning instability are you having?
Just a string or two going out?
All strings going flat, or all going sharp?

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