I have shipped a ton of stuff over the past few months (sold lots of musical gear on ebay) and had a few expensive, delicate items shipped to me. So I have done my homework on this subject.
Fedex uses private contractors. So they have motivation to deliver as fast as possible, because they get paid based on how many deliveries they make. The downside is that they will not hesitate to leave your package someplace stupid without anyone signing for it in order to complete the delivery and move on.
UPS has regular employees as drivers who are paid the same no matter how they perform. They will throw your package around or drop it off the truck, they don’t really care if its damaged. I generally avoid shipping UPS.
USPS is the slowest, typically the cheapest, and they are also the most reliable in terms of making sure you actually sign for the package and in terms of making sure it doesn’t get damaged.