It's a good question, and one I wish more newcomers to online selling considered with your seriousness.
I favor rigid foam insulation between inner and outer boxes, or at least some kind of material that cannot migrate en masse, leaving voids. Ditto to Viridian's remarks about decoupling the inner from the outer boxes. That said, I would judge the single most common packing error to be this: box too large/packing material too loose. A heavy TT, amp, or speakers can easily shift around amidst loose packing material, rendering the material useless.
When in doubt, use more packaging and fit everything snugly.
I've found that with the confidence experience brings, I can tell if a buyer's concerns are legitimate. Combined with the knowledge that your packing job was competent, a simple phone call will establish if your buyer is dealing in good faith.
Generally speaking, Fedex shipments receive gentler handling than UPS (lower overall volume), but note that Fedex Ground is often handled by low-pay local subcontractors, or worse, employees of those subs. Expect a corresponding quality of service.
I have happily shipped many high-value items through the USPS, but I admit that, with credit to careful packing, I've never experienced what a damage claim with the Post Office might be like.
Others tell me that claims can be a nightmare regardless of who the shipping agent is. Better to avoid them entirely, if possible.
