The FAA said in a memo that cameras like GoPro can be mounted eternally on airplanes as a "minor alteration" that does not require mountains of paperwork.

According to the memo, "major alterations are those that have an appreciable effect on the weight, balance, structural strength, performance, powerplant operation, flight characteristics, or other qualities affecting airworthiness".

Not likely with a small digital camera.

The memo goes on to state that "the use of suction cups, or other temporary methods of attachment (not including permanent mechanical attachments to the aircraft), would not be considered a modification to the aircraft under 14 CFR Part 43".

All that with the caveat that if the camera falls off the airplane during flight, the pilot may be charged with "careless operation", so be sure it's very firmly attached.

Happy flying and filming !