First off if we're talking Line6 and different voicings check out their Variax. You could get their 300 model (I think) for less than the strat and LP copies and still have fun with semis, teles and sitar emulations.
LP look out for poor tuners, even real Gibsons have trouble, it's the headstock angle apparently. LP look out for cracks around the neck pocket. On copies lookout for bridge pieces that don't intonate right.
Strat watchout for poor tremeloes that put everything out of tune in a few whammy waggles. Look out for selector switches that only have 3 positions not the classic 3 + 2 'inbetween' out of phase positions.
Both look out for loose jack sockets that crackle with the plug in. Look out for volume and tone controls that crackle when moved. Assistants may tell you tone knob should be on full treble but work it down and see how good things sound. Halfway to three quarters should be jazzy or alt.rock not horrible mud sound.
Take a tuner with you, get all strings perfect (beware if the Strat fights this, poor trem springs ahoy) and then give it a good play through. Is it still in tune? Les Pauls often won't be, but there are good and bad all the same. If tuning appears to be jerky then nut could be dry or sticky. If you have time tune perfectly then play the 12th fret which should be the same note, not sharp or flat. This is intonation and can be adjusted but does the bridge have enough movement back or forward left?
Action - do you get string buzzes that are audible when playing. Are the strings so high they are like cheesegraters? Are the frets worn (often on treble E and B string side). If you can play each note at each fret on each string, do any sound noticeabley quieter than the rest? Could be a dead spot...
Shops responses to you checking is always interesting. Some don't care or don't know, others will be keen to lubricate the dry nut, correct the intonation and generally remove obstacles. At the very least your show of savvy may get a discount ;-)
'The sound of the city seems to disappear'
