Packages can get damaged by anyone - the difference with the bigger international names is that damage, loss, or theft is less likely, delivery dates are reliable, and they offer full insurance and tracking, which is important. In the music store I used to work for we only used UPS for shipping instruments. Fedex (or possibly DHL) may have been equally fine but UPS had a depot in the same business park, which was an advantage for us.
Local/national postal services are more likely to lose or damage your guitar and the insurance payouts may be very limited. (Fine or smaller part or acessories but nothing expensive, bulky or delicate). Obviously with a lot of the guitar deals on this forum from the US to somewhere else we frequently see horror stories about USPS, some of which will have you shaking your head in disbelief. I would never choose them, and once the package arrives in the destination country it is handled by the local postal service, which won't be any better. (In France that is Chronopost and in the music store we never used them for anything valuable or breakable).
Whoever you choose, follow their recommendations for packing your instruments to the letter. If there is any damage in shipping, any failure to follow their rules may result in reduced insurance payout, or even no payout at all.
As I said at the beginning, no one shipping company has a perfect record. Also remember than any individual bad customer experience may not be representative of a shipping company's reliability and we have no central data collection to give real figures. BUT, I did work for a store that shipped 4000 guitars per year, to say nothing of other instruments, and audio equipment) and their decisions were in part based on past experience. In the year I worked there, we had very few damaged instruments returned because of shipping damage. In fact, the only one I distinctly remember was not UPS (a customer returned a faulty instrument via chronopost, who destroyed it). The bottom line is this - even tough I cannot give you any numbers or percentages the cheaper the service the more likely you are to have a problem and less recourse if there is a problem.