My good friend was having difficulty keeping his Les Paul in tune, so I thought I'd share these tips that really helped me with my Heritage H-150 (Les Paul style guitar). Most of these tips would work with other types of guitars. On the other hand, it wouldn't make sense to do Steps 1 or 2 for a guitar with a Floyd Rose style bridge.
I'll just add some more details and reasoning about each of the 3 steps.
Step 1: Lubricate the nut and saddles.
When you bend strings, the strings slightly slide back and forth on the nut and the saddles. If these areas aren't lubricated, then the string will get snagged and your string will get stuck out of tune instead of slipping back to it's original spot. Between the dry lubricant (like pencil graphite) or Vaseline-style options (like the Lubrikit used in the video), I don't know of any specific advantages. My guitar tech guru will attest to this.
**Tip: If you hear a plinking sound when tuning your guitar, that's the sound of your string getting snagged. Lubricate the nut and saddles!
Step 2: Tighten the tuners.
This may make the biggest difference. If the tuners are too loose, then they can easily be twisted out of tune when you're doing big bends. Be careful about overtightening and also stripping the small screws.
Step 3: Stretch your new strings.
New strings need to get set at the tuners and the bridge. To speed up this process, follow the stretching technique that I demonstrated in the video. These are the exact steps: tune your guitar, stretch the strings, and repeat until the guitar stays in tune after the stretching.
I hope this helps you enjoy your Les Paul a little more!
Watch the Video