Brompton owners

I have taken Brompton's on flights using soft bags with the bike wrapped in various things (including my clothes) to cushion them, also both versions of B&W's folding bike cases.

1. Soft bags - don't. I was lucky that it was just the EZ wheels that were broken. None of the commercial soft-but-padded bike bags are good enough for the tough treatment inflicted by airline baggage handling.
   
2. B&W Foldon case - awesome. All you have to do is remove the folding pedal, fits snugly around the bike but has some space around it where you could fit some clothes or other things. Looks just like regular luggage, so we've never had awkward questions about it from the check-in staff. Small enough that it doesn't get diverted to Oversized Luggage. Only drawback for us is that our flat isn't huge, so at home it sits on top of the wardrobe holding other things and we had to think twice about how much luggage we would own and how flexible it needed to be. Helps that it's reasonable to use it as a regular luggage case if you aren't travelling with the bike.
   
3. B&W foldon Box S - tried this because I didn't want another big case taking up space. The sides of this fold so the whole thing collapses to take up less space. Works well enough but I don't know how many journeys the sides will take. Also, it's big enough that it becomes Oversized Luggage and I have to persuade the airline staff that it's not a bike box (it isn't just one, my clothes go in there, but extra charges can become complicated).
   
   

So when we travel by air and take the Bromptons, one is in the Foldon case and the other is in the Box S.

Whatever you use, I recommend getting one of those handy hanging scales you can use to weigh luggage. When we travel with the bikes, I find myself having to do careful moving of clothes between the bags so that they all fit the individual limits as well as not going over the total weight allowance.