Clean, crisp rifling would be the first thing. Look at the date of manufacture for the barrel too. You can see it under the op rod. Mid-50's barrels are usually good barrels that didn't get shot much.
If there's a 'VAR', it's a Danish barrel that is very close to a match grade barrel. No rust or pitting anywhere. Look under the stock too, if you can. Plus the general overall condition. Worn parkerizing, etc. Dents on the stock aren't a big deal. They can be steamed out with no fuss. Gouges, cracks, etc. can't.
If the parts have matching numbers or numbers that match the S/N, run. They never had either. Part numbers are drawing numbers that have nothing to do with the S/N. It's nice if the part maker is the same, but most of those have been made that way by somebody with too much time and money.
"...specific problems with..." Nope. Some are more rare than others though. Springfield Armory Inc. made some M1 copies. That'll be stamped on the receiver. They're ok, but pricey.
"....30-06 vs .308..." Pretty much a preference thing. However, some .308/7.62 barrels don't have the right diameter gas hole. Should be around .080". Causes unreliable functioning. Usually depends on what .308 barrel was put on and by who.
Do not buy a rifle that was 'converted' by installing a chamber insert. They tend to come out with no warning upon extraction. Isn't particularly dangerous, but you'll have some odd looking empties and be back to .30-06. Likely with a rough chamber.
Where you find one these days is luck of the draw. Gun shows can be good, but I've seen junkers at high prices at gun shows. One guy at a local show lugged a rifle, in poor condition, back and forth at lot because he was asking $775 for a beat up rifle. Same guy wanted $4 each for clips.
