Friction Point 1: the nut
Most guitar tuning instability can be traced to the nut. The strings pass through narrow grooves with a lot of downward pressure, so they have a tendency to stick and bind in the slots. Nuts are often pre-slotted (or slotted in a hurry on a production line).
Making a nut work smoothly requires time and effort, but it’s something you can do yourself. If you notice pops or clicking sounds from the headstock when you bend strings or use your trem, the nut may need attention.
Nuts are often left too high above the string slots, so even if the depth of the slot is correct, the string sits too deep (see pic, above). The top of the nut only needs to be high enough to hold the string securely in the slot, so in effect the top half of the string can be proud of the top of the nut. This helps keep the contact area between the string and the nut to a minimum, thus minimising friction (see below).
Nut files can leave grooves in nut slots. Once the depth of the slot is optimised, the inside surfaces of the slot can be smoothed out with wet and dry paper. Try 1200 or 1500 grit, maybe wrapping the paper around cut-off lengths of string. You can follow up with some polishing compound; we find that chrome polish is particularly effective on bone.
2. Critical But Stable - Tremolo Workshop