To my knowledge there are no studies correlating ANR (or good quality passive) headsets with an increase in pilot error, but a several have been done on noise fatigue. There's even an old FAA advisory circular on the subject which basically boils down to "Multi-hour flights listening to the engines drone on and on will make the pilot cranky and error-prone" - so from that perspective any good-quality headset is a safety enhancement.
As far as missing auditory cues because of ANR (things like the stall warning, funny engine noises, etc.) it's generally not a concern: ANR headsets are good, but they're not THAT good. Because of the way ANR works sudden sounds will cut through. You'll miss "long-term noise" (like the beat frequencies Falk mentioned from unsynchronized propellers), but you're not likely to miss anything safety-critical.
(One notable exception: ANR headsets are good at cutting out wind noise, so if you have a partially unlatched door you won't hear that.)
What you will notice with ANR headsets is that when the battery dies things get loud - sometimes louder than cheap passive headsets. Carrying extra batteries is a must with ANR headsets since having your noise reduction go away in the middle of a flight would be somewhat unpleasant.
As far as music, that's a mixed bag. Much like having a CD playing in your car, I don't believe it's a significant safety impact unless you're being irresponsible with it (blasting it at a volume that drowns out everything around you, or rocking out when you should be watching for traffic).
Additionally, all the intercoms and headsets I've seen give the com radio priority, and will either cut out or substantially mute the music input when there's radio traffic. This makes you less likely to miss radio calls, but it also makes listening to music in the plane an absolutely DREADFUL experience (in the northeast where I am there's always somebody talking on the frequency - if you hear half of the song between the radio cutting in it's a good day).
I personally find the distraction from the music muting/unmuting to be more of an issue than boredom from not having music in the first place (if you're that bored you can always look out the window, or tune a local CTAF frequency and listen to the antics), but your experience may vary substantially if you're in an area where the radio is quieter.
Policies for commercial pilots would vary depending on the operation, but for airline pilots using your personal cell phone is forbidden, and before it was forbidden by the FAA it was usually prohibited by the airline.
Part 91 operations (a ferry flight, or a crop duster spraying the local fields) wouldn't be subject to that restriction though, and making a phone call or listening to music would be permissible under the FARs. With all the studies on the negative safety effects of using a cell phone while driving I wouldn't recommend the phone call, but I imagine crop dusting is more fun if you have some music to listen to…
