I have a padded soft bag that I used to take my bike(s) on long haul flights all the time. It used to be that bikes were classed as sporting equipment and taken for free on BA (back in the day when you could only join BAEC with a full fair ticket etc.). The bag is smaller than the ones usually found in bike shops, which makes it easier to get in and out of cars, but with no wheels a PITA to carry around - but I had carried it on the Underground, it just is hard work, but the beer at the end of your journey will be well earned!

For my mountain bike(s) I used to have a fork spacers (get one from a bike shop, they used in the packing of new bikes), and for the rear a piece of plumbing polly pipe cut down for the rear drop outs with a quick release. Take off the rear mech and zip tie to the polly pipe. The frame use foam to protect (such as insulation for water pipes), unbolt the handlebar and zip tie it to the forks and throw in a load of bubble wrap, wheels each side of the frame, and clothes into the bag for extra protection and to pad it out.