Well, it's certainly impossible to have these shipping companies to be 100% free of any damage of some sort. Every one of them has had numerous packages go missing, damaged, delivered to the wrong place, etc. Chances are, there is a high enough % that your package will arrive safely to the recipient if packaged properly of course.
But anyways, I have been using the USPS the most and 99% of the time, everything went well. There was just that one incident I had years ago where the substitute USPS delivery person must've either delivered a package of mine to the wrong place and or it really went lost somewhere. Tracking showed it was delivered, but here was the kicker, the seller did not do signature confirmation on it!?! I was pissed especially on something of this magnitude. Pretty bad because it had about $10k worth of golden age books in them.. Gladly I got my $ back from the seller. Made a post of this way back then.
Fed-Ex Ive used a few times and they have the best tracking system. I sent high dollar books through them and no problems at all.
UPS I've sent a few cheaper items, again no problems either for me.