Fedex - They never pay claims, Our company is done with them
As a major shipper that has shipped countless thousands of packages with Fedex, I can say for certain that they have a policy of never paying out a claim unless they have to.
(They have NEVER paid a claim in my experience no matter how badly they screwed everything up.. I assume that legally they have to pay a claim to someone... but I have never seen it)
If you are shipping worthless junk, they are fine. Otherwise use USPS. Which does have decent customer service, and they DO pay claims then they mess up. They get the packages where they need to go I guess, but not always intact, and they never take responsibility when they do crush or destroy it.