Re: Best way to go about buying a LP
I have eleven Les Pauls.
First, set a budget.
Play a lot of guitars.
Decide which neck profile you like. They all vary, but basically you're going to be choosing between a '60 Slim Taper profile or a rounded '50s profile. Historic models have a bit more variation, based on the year being reissued. I know my SP Supremes have a different profile than any of my others (both Historics and production models--'60s and "50s necks), and I think the Customs have a similar profile to my Supreme.
Play a lot of guitars.
I have five chambered Les Pauls. If weight is an issue, then by all means get a chambered guitar--it is still a Les Paul. All of mine sound excellent. Some guitars have weight-relief holes (swiss cheese) cut into the mahogany under the maple cap; Historic models come in chambered versions, but they specialize in totally solid guitars make from billets of select lightweight mahogany. There are several design and construction differences from the Historic models opposed to the regular Standard, Traditional, Custom, Classic models. They are SIGNIFICANT, and priced accordingly. Oddly, all of my LPs weigh in between 8.9 and 9.15 lbs.--except two chambered 2007 Classic Antiques (discontinued) from the Guitar Of The Week series that come in at 8.2 and 7.9 lbs. They feel like toys compared to some of the others, but they totally rock.
Play a lot of guitars.
Like some of the posters above, I also look for a loud and resonant guitar. I like to strum the open strings hard and feel for the resonance at the top of the head stock. I look for a guitar that just vibrates madly.
Play a lot of guitars.
Look for something that just feels "special" in your hands.
Play a lot of guitars.
You'll need to make some decisions on cosmetics. Do you like the look of a Custom? Do you want a top with a lot of quilt or flame figure? Expect to pay more for those than a plain top. Can you live without neck or body binding? A Studio Faded might be all you need. I often tell people to buy the most, nicest flame they can afford.
Play a lot of guitars.
Do your homework. There are several good books on the Les Paul guitar, Tony Bacon's is good, Iwanade's "Beauty Of The Burst" is a MUST HAVE; there are others. Hang out on the Les Paul Forum. Ask a lot of questions. Know the differences between a short tenon and a long tenon; a Nashville bridge and an ABR-1. Check out some of the websites, like Gibson's, the Music Zoo, Private Reserve, Wildwood Guitars, Dave's Guitar Shop, et al. Knowledge is power.
Play a lot of guitars.
Always good to have a lot of cash on hand when you are guitar shopping. Be patient, be stealthy. Strike hard and fast when the prey is in range. Bargain, bargain, bargain. Cash talks, BS walks.
Play a lot of guitars.
You can find some good deals used, but "caveat emptor". You must know what you are looking at and what your are looking for. You don't want to pay $1600 for a $350 Chinese knock-off. Know how to tell the difference. Often, you can tell simply by the smell. Seriously. I have doubts about buying a used Paul off E-bay. I have bought some of my G&L Legacys there, but LPs...they are different. I suggest developing a good relationship with your local dealer. The occasional six-pack of micro-brew or a batch of fresh homemade chocolate chip cookies can get you in good standing, lol!
Play a lot of guitars.
If you are desperate for a good deal, don't rule out a guitar with a broken headstock. It's a common occurrence with Gibsons. You may be able to find a great guitar at substantial savings. I have three Gibsons that I bought with headstock repairs--all are stable, and excellent sounding and playing guitars. They were either damaged in shipping or damaged in the store. The finishes were touched amp and the repairs are barely visible. Another LP Historic I bought has slight cosmetic damage from a fall in the store--I bought it for cost. But I do have one rule about these guitars--I will not buy one that has ANY damage to the headstock faceplate. No breaks or even a crease through the faceplate. Damage to the faceplate can be repaired but it is expensive to correct and have the cosmetics right. If you can buy it cheap enough, a guitar with a broken headstock can be a real bargain. AND, a great player.
Play a lot of guitars.
I use Schaller Strap Locks on all my Gibsons--the stock strap buttons are a disaster waiting to happen. Parts can be changed out on a Les Paul--pots, bridges, pickups, tuners, tailpieces--all can be customized. But you're not only looking for the prettiest girl at the dance; you want the one that has that inner beauty, a soul, a loving heart. You can't make a silk purse...etc. Learn how to distinguish between a dead guitar, versus merely a guitar with dead strings. Do budget for a professional set-up. And the trick for getting a great playing LP is making sure that the luthier uses a radius gauge and files the saddles to set the radius of the strings at the bridge to match the radius of the fretboard. Usually this is a 12" radius on a LP. Usually.
Play
A LOT of guitars!!!
That's my advice--you may have noticed a common thread.

My favorite LPs are my Historic models; I have an R9, an R8 and two the Guitar Center G0 versions--one of those G0s in particular is awesome/great/to die for/the bee's knees. etc. I prefer the rosewood "Standards" over the Customs--over the years, I've never really bonded with a Custom, and they tend to be a little heavier than some of the other models. YMMV. I do like my Supremes though. They benefitted from adding Duncan Seth Lovers, RS Kits and Gotoh LW aluminum tailpieces.
Drool time. Here's a pic of four of mine. From left to right:
2005 '58 Historic Reissue, R8, full Gloss in Washed Cherry, stock, solid body, huge neck, ABR-1 bridge,stock BB 1 & 2;
1996 Classic 1960 Premium Plus in HCSB, weight-relieved, '60s Slim Taper neck, ABR-1, Duncan 50th Anniversary A2 Seth Lovers, RS Kit, LW TP, reflector knobs, aka "Lolita Paul";
2006 LP Supreme in Heritage Cherry Sunburst, maple cap front and back--no control panel on the rear, chambered, Custom neck profile, Nashville bridge, LW Gotoh TP, stock 490r/498t pickups (I hate these);
2007 Guitar Of The Week "Tom Morgan Artist" chambered Classic Antique, all stock, very light weight, '60's Slim Taper neck (smaller and flatter than Lolita), Nashville, '57 Classic/Classic +.
The 'R8 and Lolita:
The Supreme: (I have a second one in the Root Beer finish.)
The girl with her lipstick on crooked:
Happy hunting!
Bill