i employ the following checks when buying guitars, Les Paul or otherwise:
*Headstock:
-check for broken units/ cracks. an attempt to hide this would be a refinish job.
-check for functional tuners
*Neck:
-run you fingers along the fretboard register, make sure no frets are jutting out
-check for frayed neck binding (if applicable)
-check for worn out frets
-buzzing- check for headstock/ nut/ fret buzzing. as long as this doesn't get amplified- it's good
-sight the neck, make sure relief isn't too excessive
*Body:
-the usual suspects: cracks/ dents/ scratches
*Electronics (if you are testing through amp...)
-make sure ALL switches/ knobs are working
-Input jack: twist your instrument cord & listen for cracklings/ hissings. if the signal disaapears while doing this- it's bad... (make sure it's not the fault of the cable, yes?)