Here is a quick inventory of some show stoppers:

• Cracked frames - usually near the bottom bracket area.
• Overtightened seatpost bolt cracks frame.
• Frozen stems and seatposts - corroded shut into the frame. These may take extreme measures to remove - over several days. This may involve the use of noxious chemicals, a blowtorch and large tools.
• Stripped steerer-tube threads.
• Stripped or cross-threaded BB threads. BB may be non-replaceable.
• Bent fork or other frame alignment issues. Was the bike evidence in a vehicular homicide case?
• Evidence of a big frontal impact - ripples or cracks in paint in top tube.

Following is a list of things that may not be show-stoppers, but factors in your purchase. I've worked on hundreds of used bikes. Almost every one had a worn out chain and more than half had a worn out cassette/freewheel and chainrings. Factor in the cost of replacement drivetrain parts in the purchase price.

• Many had indexed headsets, some were in such a bad state that I wonder how the previous rider could actually steer the bike.
• Most hub and bottom bracket bearings needed servicing. Many needed outright replacement.
• Many had bent derailleur hangers. Some on alu frames with non-replaceable dropouts. Most kids derailleur-based bikes had mangled rear derailleurs and bent dropouts.
• Most needed derailleur adjustments. Based on hard experience, I know now that your average bike rider accepts a drivetrain that randomly jump between gears and noisily clatters along during riding as being 'normal'.
• Most needed new shifter and brake cable housing.
• Many had rims worn out from braking, and most needed some wheel truing. Many rear wheels had unrepairable hops due to impacts with curbs.
• The last used bike I bought had a frozen pedal. After several days of extreme measures including a 300 pound mechanic jumping up and down on a 15 inch wrench with a cheater bar, we had to give up on the crankset. Too bad, because it was a $400 carbon crankset.
• And about 20 other potential disasters in the making.

Good luck! I've obtained some outstanding deals on used bikes, but most used bikes have 'issues' and the sale price should reflect this. Many of the bikes I see are outright death traps.