I have used UPS, FedEx, and USPS.
USPS is the cheapest and I've never had a problem with them. But, their tracking info leaves something to be desired.
Actually, I have never had a problem with any of them. But, others have posted on here that UPS has bought more basses than anybody (i.e. they lose or damage more).
I have almost always shipped basses inside a hardshell case, inside a cardboard box.
I have shipped USPS for as low as $22 or so.
Mostly, I have used FedEx ground and even shipping FL to CA usually only runs around $30 - 35.
Recently, I setup a personal account with FedEx and I measure and weigh the box at home, then go online and use my FedEx account to generate and pay for a shipping label. Print it out and take the package with the label to the nearest FedEx/Kinko's Copy Center and drop it off.
Doing it using my personal FedEx account has been the cheapest of anything except USPS.
At the other end of the spectrum, I had a buddy of mine in Chicago buy a bass for me and ship it to me. He took it, in the hardshell case, to a UPS Store and paid them to box it up and ship it to me (insured for $500). That ended up costing me $80! The UPS Store is a total rip-off for packaging. AND, they charge WAY more for insurance than it costs if you do your own mailing label online, for UPS.