I've only ever used Fedex, out of convenience. Out of about ten shipments over the last year and a half, they have broken two but I blame my packaging, not their handling. I read somewhere to preprint labels to avoid the conversation they want to have over packages marked fragile and I think that has helped. Every package where I've preprinted the labels, they treat it more like a drop off and don't seem to care about contents and reasons for the fragile label.
I agree with pete b: pack it so doesn't matter.
I've shipped for bottle shares not competitions. Since I am sending out bombers to friends, I invested a bit in a 6 pack of U-Line wine bottle shippers. I add a little bit of brown paper or bubble wrap to fill the extra space. None of my shipments using these shippers has been a problem--both packages that ended with broken bottles were hand packed with other materials.
I have also had success re-using the packing material my Noble Brewer subscription comes in. I have had one breakage re-using those shippers, which hold four bombers very neatly. In that case, the packaging was damaged on receipt, even though the bottles were intact. I tried to shore it up with packing tape and bubble wrap, a mistake I won't repeat.
I am not sure I'd recommend the U-Line shippers for competitions, though, due to the added expense. When doing bottle shares, I ask my friends to send them back, even empty, since they stand up very well to re-use.