We've had decent luck shipping bikes (overseas primarily). We have a good strong case (Competitive Cyclist is where we bought it). Personally I would not trust my bike in a cardboard case...I would just worry about it.

Flying internationally though, although you can check as luggage, they generally charge an add'l fee for the bike. Very few airlines will do it free anymore. Ours was I think $50 or $75 last trip on KLM from Netherlands to Chicago (each way). Each airline has a section on their website as to charges.

I agree, shipping with UPS and Fedex is about the safest I think. Our last time shipping skis and bikes with Fedex was a very good one. You can pre-ship and make sure it's there when you arrive so it works out well and now that airlines are charging anyway for the bikes packed in the sturdy / larger cases, it's worth the few more $$. (No lugging it at the airport either!)