Adding to Reirab and Brian Towers' superb answers on this topic:
There are several ways an electronic device can interfere with aircraft operations, but the likelihood depends on age, aircraft design factors, and the nature of the EMI aggression (frequency, modulation type, and power -- there are also broadband EMI sources on aircraft, but we can safely disregard these for the purposes of this discussion, as PEDs generally are narrowband emitters unlike, say, an electric hydraulic pump which lost its interference suppression capacitors, spewing broadband hash as a result).
The typical concern, though, is of interference with an aircraft's sensitive navigation (ILS/LOC, G/S, NDB, GPS/WAAS, radio altimeter) and communication (HF, VHF, SATCOM) receivers (it's possible to interfere with the transponder/TCAS and weather radar signals as well, but those aren't quite the same level of concern as say the glideslope is), from intentionally emitting PEDs -- cell phones (especially GSM phones, as the GSM air interface is more EMI-prone than that a CDMA phone is) and 2.4GHz or 5GHz devices (WLAN, BT) are the main concerns on this front, although unintentional emitters (anything in airplane mode) are also a source of potential trouble.
Furthermore, comms interference takes an immediately noticeable form -- "what's that funny noise in my headset?" (Pilots of small General Aviation aircraft have reported this on a regular basis.) This is in contrast with interference to navigational receivers, which may be subtle but not immediately troublesome (losing a bird in a large visible GNSS constellation), obvious (a fail flag on a VOR or LOC), or insidious (a false glideslope indication).
However, the consequences are high here, especially during coupled or autoland approaches -- a human pilot can see that a full fly-down indication appearing out of the blue at 300' RA during an otherwise stable ILS approach is bogus and whack the TOGA button to get the crew some time to tell the noisemaker to zip their electronic lips, but an autoland computer won't know the difference and will follow it blindly.
A good piece of news is that in the somewhat-more-common case of an unintentional emitter (a device in airplane mode, for instance), the emissions tend to be simple, unmodulated carriers -- I see this as more likely to trigger a FAIL flag than a flight path deviation. (I will put another question up on this topic, even -- it's worth asking about, and perhaps even an experiment or three.)
However, due to the large number of airline flights yearly (66 million according to ACI as per reirab's comment below), something that happens once in every 10^7 landings (Extremely Remote according to AC 25.1309) will happen every other month on average, and even something that happens once in 10^9 landings (Extremely Improbable according to AC 25.1309) is bound to happen every 15 years or so! The reported PED interference instances are not nearly as severe in consequences as the 'worst case' of a full 'fly down' during the late phase of a coupled or autoland approach (or false radio altimeter readings causing a grossly premature mode shift during autoland), though, but the probability of PED EMI with aircraft systems on any given flight is enough to make it a concern -- 75 events were reported worldwide to the IATA from 2003 to 2010, and given this number, we get a probability of roughly 1 in 10^7 for PED interference on a flight.
From this, we can conclude that it would be highly unwise to take the chance of having a device talk all over the ILS signals during a critical phase of flight -- you would have to find a way to reduce the probability of PED interference with aircraft systems to 1 in 10^11 or better (this is 100 times less likely than what is currently allowed for a catastrophic event) in order to have a chance of challenging this wisdom.