Underneath, as mentioned. I'll add to check the rear torsion bar anchors @ the trans crossmember, and the top of the frame rails at the upper control arm mounts.
Puddles of mystery fluid anywhere on the guy's driveway or garage floor. Might have been this car, maybe not...
Dry brake drums/backing plates, especially at the bottoms. Could be brake fluid, could be axle lube if they're wet. Check for a solid feeling brake pedal- nothing "mushy" or "slowly sinking"...
Look over the wiring harness for excessive electrical tape or crimp connectors. Sixty year old cars have had a lot of hands on them.
Missing parts of ANY kind.
You didn't say how it's being presented- as a restored car, as a project, or something in between. Anything that doesn't strike you as being what you were told is going to be just the tip of the iceberg.
Ask to see receipts or pictures of any claimed work that was done.
Take a magnet, wrapped in a chamois (so you don't scratch paint). Check the bodywork, especially at the front doglegs and behind the rear wheels.
Most of all, I don't want to see you on here later saying "I took his word, he SEEMED like an honest guy"... NEVER assume anything about the person or how the work was done. Look for/ask for proof. If work was done by a shop, ask for names and references and check them out.
And if you're interested, make what you think is a fair offer. Don't lowball the guy, or pay an exorbitantly inflated price. Be fair, but be prepared to walk away if necessary.