Like any old car, rust. PV's are more secretive about their rust, and it tends to be more structurally significant when it occurs. Check the following places:
1) The front frame spars - PV's have a mildly unusual front structure, two straight rigid spars extending forward from the firewall to behind the bumpers. It's the only structural element in the front end. They 'Y' at the firewall, with one arm going up and the other end extending under the front floorboards. The lower arm is prone to rusting, and when it finally lets go the whole spar gets wobbly. The front end of the spar is rust prone too, dropping off the bumper, the sway bar, and eventually the idler arm/steering box as the rust progresses towards the rear.

2) The driveshaft tunnel - specifically the rear end of it. This is more of a structural element in a PV than in later cars. 4 out of the 5 rear suspension links all come together and mount to the car at the rear end of the tunnel in a pretty small area. Which is prone to rust and can result in the axle coming loose.

3) Rear spring perch subframing - there is a lot less subframing on a PV than in later cars (probably why they weigh 300-ish lbs less). One small snippet of subframing that is (supposed to be) there are arches inside the rear wheel wells, from the rear floorboards, over the 'hump' and partway back to the rear bumper. Thse are prone to rusting, and eventually result in the springs shooting into the trunk area.

And, of course, some less critical but 'leading indicator' rust areas:
1) The rear section of the trunk floor. Recessed, prone to taking on any water that leaks through the trunk seal, under rubber mat so the water can work on the metal for a long time.

2) As on all old cars - the floorboards.

3) As on all old cars - the sills.
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'63 PV544 rat rod, '93 Classic #1141 245 +t