My bike goes in its S & S case. My wife's goes in a Bikeflights cardboard box for now. Just recently became aware of this thing:
https://postcarry.co/ thanks to Russ at pathlesspeddled. If it seems like my wife's bike is going to make many more trips, that might be worth having. The biggest hassle on our last trip was, after landing and assembling our bikes, realizing that we had no good way to carry the bike box out of the airport. That backpack feature on the Post Carry box would have been great.