I’ve only played one Les Paul (amplified anyway) and it’s a Studio I’ve had for almost 30 years. Since it’s a keeper I’ve always played other makes and models when trying things out.
To answer the question though, I’m going to say whichever of his LPs Duane is playing when they break into Soul Serenade here:
Some other leas conventional (but still famous) favorites are Gilmour’s Gold Top on Another Brick in the Wall pt. 2, and Neil Young’s Old Black. Probably Cortez the Killer, or On the Beach, or even Big Time, or his ACL version of Down By the River:
And RIP Peter Green. Can’t mention Les Pauls today without mentioning him.
back when I was in LA on business all the time, the music studio I was working with lent me a guitar to play at my hotel (I was there on an 8 week forced march kind of project completion). The guitar was a ratted out 69 LP Special, it was a relic before relics were a thing. It played very well, held tune well also. It was an old Road Dog, the total opposite of a "Case Queen", but it worked for me; not getting out of practice and most of all not letting the callous' go away in that two month stay.
Best sounding LP I've played was a maple bodied 54 Junior.
Best sounding LP I've owned was the Epi gold top I had. The Duncan Antiquities and 50s wiring beat what came in all the Gibsons I had. But the 2014 Gibson LPJ was a close second. Interestingly, those were the two least expensive Les Pauls I had.
My favorite recorded LP tone is the gold top Deluxe with P90s that Mike Ness plays in Social Distortion.
You can dial in that LP syrupy sound with a lot of LP type guitars. The real question is which amp sound do you like hearing it though? Which mic captured that perfect sound? What did the engineer dial in with it so it had the perfection? Because it all came together.
Best sounding Les Paul I've played was a 1960 custom with the three PAF's, the bridge pickup had that fat rounded highend that was incredibly expressive. I just get lost playing a guitar with that sound.
Sticking with what I've played: A Norlin era Deluxe through my early model 50 watt JCM 800 into a Celestion loaded Sonic 4x12. Those older low output minis just sounded so good in a rock sort of way with that rig. Clarity and overdrive!
That same Norlin Deluxe also sounded really good through a Twin that I had back then.
I also had a Standard at the time, and I've had a bunch since, but that Deluxe remains in first place for the Les Pauls that I've owned.