As this is topic where people tend to tell their own opinion instead of facts and research results, we should anatomize the myths first. As some of the answers already stated correctly, I will repeat - airliners are shielded from such interference. Aircraft must stand electrical storms, microwave energy, cosmic radiation and all sort of signals. Never any incident nor accident related to that, and we know people use portable electronic devices (PED) all the time when they shouldn't, video recording, texting etc. Calls are sometimes picked up by certain antennas installed on the aircraft and you can hear the signal via headset. How about WiFi and interactive PTV's? No interference found in normal operations as signals are at totally different levels. Localizer, glide slope/DME, TCAS, VOR, everything use different transmission levels. Don't believe - go to cockpit and call home, see for yourself. Multiple tests by power, radiation and frequency were run by both official and independent agencies, including airliners, Boeing, Airbus, FAA and so on. I have actually read many of the results and tested it myself when possible. I encourage you to do the same. Some non-US airlines not only allow but actually support calling and texting from the air ($$$).
Few times there were cases of A/P disconnection, false caution alarm, course deviation and other random strange occurrence in flight. Sometimes the electronic devices from the passengers were to blame. Boeing bought them, brought them to labs, set aircraft systems to flying configuration and observed. They would fail to recreate the interference every time, except the obvious ones, that wouldn't happen in real life anyway. Note that most researches on the subjects are done years ago. Today PED are smaller, safer, use less energy and emit less radiation in addition to all fancy electronics installed on board.
Nevertheless, there are several good reasons why the use is prohibited, and should stay prohibited in the critical flight segments.
First, passengers should listen to safety instructions and not play Angry birds.
Second, when taking off, they should be aware of environment in case anything goes wrong.
Third, anything loose is a possible missile in the airplane, PED included. That's why flight attendants will ask you to put everything not on you in the overhead bins.
Interference is always possible. It is not completely disregarded.
- at landing and taking off more sensitive equipment is used which could be distracted by large electromagnetic radiation
- emergency and backup systems emit on the similar frequency as common PED, they should be ready to use at taking off or landing (more than at FL370)
- older aircraft - older equipment - much greater risks, especially old GA aircraft. PED avoidance recommended. PED interference proven to be existent, especially with XHF and ADF.
- GPS signal interference is recreated in labs after intensive high-power output. Not likely to happen in normal, commercial operations. Otherwise we would watch it on the news every day. Remember, for a serious problem to arise in a flight, history of air accidents tells us there has to be a critical chain of events, not just one cell phone call.
In the US, FAA's stance is "better safe than sorry". And that's good. For now.
Note: It is much simpler and cheaper just to stick to (valid) recommendations than to test each and every existent portable electronic device on the market with each and every existent aircraft on the market in each and every segment of flight. To allow something like PED, you have to prove it is safe to use, rather than anything else. Who has the money and time to deal with such "trivial" matters?
The bottom line - don't use PED if told not to, apart from safety redundancy/margins, there is always a possibility signal may be picked up. And possibilities are taken very seriously in this business.