I know most of the things I'm gonna say are already well known, but I just want to give some first-hand experience so as to erase any doubt.

I usually practice on a rather cheap Les Paul copy that I upgraded somewhat (pickups mostly), but as with cheap guitars, there were tuning issues.

However, those went away almost completely once I changed the nut to a Graph Tech. Such a deceptively simple upgrade, but the improvement in tuning stability was dramatic. Not a little, not a lot, dramatic. Where before with the cheap plastic nut the thing would go out of tune every couple of songs or so, with the new nut I haven't touched the tuning keys in 3 days or so. I am simply amazed at how such a small upgrade could make such a huge difference.

So... if you're thinking of doing this upgrade, I wholeheartedly recommend it.

The nut is this one .

Also it REALLY helps if you string your guitar as described here. (that's not me)

And give the strings a good stretch.

Anyway, that's about it. Ignore it, close it or post other ways of improving tuning stability if you want.
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Yes, but only in a way indistinguishable from random luck or the result of your own efforts.