The single most important thing you can do when purchasing an aircraft is to have a mechanic of _your_ choice inspect the aircraft and maintenance records. This should _not_ be the mechanic who's been maintaining the aircraft and signed off the last 15 annuals or one recommended by the current owner. It should be someone who's familiar with the model of aircraft in question. The point is that you want the unbiased opinion of someone who's looking out for YOUR interests.
Yea, this will cost some bucks, but better to spend $100 or $200 on a pre-purchase inspection than to find out about that $5000 AD after you buy the thing.
Apologies to all those who have heard this 9000 times already, but sometimes a point bears repeating.
One thing that I found out is that many Leins held by the Grumman American company (from original purchases) were never closed with the FAA (they have to register them with the FAA) because paper work got screwed up or lost when Grumman got sold. My plane had an $18,000 Lein still on it from 1976 from the original purchase. It was very difficult to track down the person in charge of this Lein. I forget the name of the company that I finally found (It still had the name Grumman in it. Something like the Grumman Loan company. I can find out by looking in some of my paper work). When I found them they acknowledged that tons of Grumman planes did not have their paper work closed with the FAA. They did it for me in one day and faxed me a copy.
Sometimes the logbooks make for interesting reading. You might find something like "Replaced nose gear strut and torque tubes and installed doubler plate on belly at station 51.0; replaced prop". No "damage history", but if you read between the lines it was pranged...
Speaking of logs, the logs should show a pattern of steady maintenance: regular oil changes, inspections (at least annuals), the usual run of wear items like tires, brake linings, filters, being replaced, and a steady diet of recurrent AD compliance (like 79-22-04, aileron inspection, on the 4-place aircraft). Any gaps are cause for investigation.
By looking at the times recorded in each maintenance entry, you can get an idea of how the aircraft has been used. If you see two annuals, a year apart, at 1200 TTSN and 1210 TTSN, you know what that aircraft has been doing during that year.
While you're at it, keep an eye out for tach/hobs replacement, as evidenced by a maintenance entry stating such, or by strange jumps in the times recorded. Just because the tach says 1200 hours it doesn't mean you're looking at a 1200 TTSN aircraft.
Rebecca suggested a title search. I second that recommendation.
I nearly got bitten because I didn't do one. I paid cash (having taken out loans on other things) for 46L. When the time came to do the engine, I took out a loan on the airplane instead. As the bank was doing the paperwork, they found an old Lien against the airplane which was from 3 owners back. It wasn't hard to clear up, as when I called the bank which held the Lien, they informed me that they did not make airplane loans and hadn't in memory done so. As a result, they had no problems filling out the paperwork to remove the Lien. You might not be so lucky.
Something else to consider...
I've heard of people who file a lien against their own airplane. That way, if it's ever stolen, the thieves will have a harder time selling it.
This does assume that the purchaser does a title search, and the thieves will want to sell it as a whole as opposed to parting the thing out.
One of the reason I also chose Grummans (aside from the handling, ease of maintenance, heck ya'll know the reasons) was that with the limited numbers, they were not as likely to be stolen as a Cessna. Most of the supplier of our parts we know, and know our planes so who would the stripped parts go to. Bit extreme if you just happen to need a part.