When I've travelled with my bicycles, I've always packed them up in a box and have taken them on the plane. I've lost count of the number of times I've done this ... 30+ times. Within North America a bunch of times, back and forth to Europe several times, back and forth to Australia several times, within Australia a bunch of times, up and through Asia ...
Only once so far have I had an issue ... a light broke. My fault, I should have removed it and packed it.
But you've got to do your research. Some airlines charge a lot, some are really good.
Delta is one of the worst for bicycle fees ... Air Canada is one of the better ones. Australian, New Zealand, and British airlines are even better.
You can read this page about the history of travelling with your bicycle on a plane ... or continue right to the chart that gives you the prices and regulations for each airline.
Airline Baggage Regulations For Bicycles
But don't depend on that chart ... look up the baggage information on the actual website for the airline.