Thanks, everyone. So the consensus seems to be that it's no more dangerous than other aircraft, though it may have comfort issues in economy class.
Comfort isn't a critical factor for me. I much prefer to fly with US-based carriers, in spite of their generally horrible service, as their safety records are usually impeccable. Unfortunately, that's not always possible where I live, and more and more local carriers are using the 787 (though not Qatar Airways, interestingly).