Hello - We were in Amsterdam and just got back. Yes there are rumors of Delta Filing Chapter 11 - mostly related to the Pilot contract issue. If you have any tickets on Delta, rest assured that you will be fine - I don't think there is any probability of Delta closing up business totally in the next few years (we have close to 1,000,000 miles with them). Airlines tend to go in and out of bankruptcy for a while before shutting down totally if then. United is either in or just came out of bankruptcy.
USAirways is about to go back in. Now, this is one airline you may want to have some concern about. Do not buy tickets for after this fall or so right now. Having said that, I recently cashed in virtually all my miles with them for award travel next February in First Class on Lufthansa.
I would not worry about any airline entering Chapter 11. Even Chapter 7 is not the end of the world since often times other airlines will honor your existing tickets. Frequent Flyer miles are likely to be taken over by another airline (eg Delta took over Pan Am's miles, Continental took over Eastern, American took over TWA's).