This article is from the Aviation FAQ, by Geoffrey G. Peck firstname.lastname@example.org with numerous contributions by others.
FAR 91.21 governs portable electronic devices. Use of a receiver is
prohibited except for units which "the operator of the aircraft has
determined will not cause interference with the navigation or
communication system of the aircraft on which it is to be used."
"(c) In the case of an aircraft operated by a holder of an air carrier
operating certificate ... the determination ... shall be made by that
operator of the aircraft on which the particular device is to be used.
In the case of other aircraft, the determination may be made by the
pilot in command or other operator of the aircraft."
In plain English, this means that on an airliner, the _airline_ must
allow you to operate the radio -- the captain does not have the sole
authority to authorize its use. On a private aircraft, the captain does
have that authority. Note that amateur (ham) radio operators are
forbidden by FCC regulations from transmitting on _any_ IFR flight.