USPS within the US, has been better than normal, until it gets to a local level, for me at least, my own PO is terrible but I understand others are very good. Good luck getting a claim fulfilled though, insurance just attracts part timers sticky fingers imo.
UPS has seemed to be the most reliable in the southeast at least, although they also don't take any liability for packages left with no signature, I would say definitly ship basses with them, and see below -
FedEx is super fast for small things, if I get a box over 12" though most likely it will show up as a ball, not a box it's so damaged.
Honestly, bass wise find a buddy that can receive freight, you'd be amazed how much less it is for oversized on a skid, and I've had much better response having packages coming to a storefront rather than residential if that's an option.
And then there's the odd ball route, anything coming or going to California only takes 2 days USPS??! it takes more time to mail to neighboring towns..