I have read that passive hearing protection (standard earmuffs/plugs) offer good protection against higher frequency noise but do not protect as well at lower frequencies. Supposedly active noise reduction (ANR) can attenuate low frequency noise. Does any hearing protection worn by professionals in aviation (airports, military roles, etc.) anywhere in the world use ANR or is it all passive?
Adding to the civilian answers, the F-22 helmet and F-35 helmet do include Active Noise Reduction technology. I believe most other US military fighter jets rely on passive noise reduction; this may be in part because ANR technology didn't really hit mainstream use until the 1980s, while many of the fighters in service with the US today (F-15, F-16, F-18, A-10, EA-6) began production in the 1970s or earlier.
Not quite on topic, but you if you don't have ANR units you can make very effective low frequency earplugs without spending a fortune on Flare Audio Isolate aluminum earplugs (I bought a set for 40 bucks and promptly lost them, although they do work). The metal core of the Isolates helps the foam earplug part attenuate low frequencies quite well. If aluminum = good, lead must = better!
So I just got some 3M Push-Ins earplugs, the kind with the blue plastic shaft and a yellow bulb, some lead foil tape used for ballast on golf clubs from a golf supply store, and some quarter inch heat shrink. If you're concerned about lead encased in heat shrink in your outer ear canal, you can also use aluminum or stainless steel foil tape for somewhat less attenuation.
Wrap a 2-3 inch length of tape around the shaft (depending how thick you want them, which depends on how wide your ear canal is), cover with heat shrink and shrink it over (I put the bulb part into a piece of metal tubing the diameter of the bulb to protect it from the heat gun).
I tested them in using stereo headphones cranked way up with low frequency sounds playing, comparing them in opposite ears against foam plugs and the Isolates (before I lost them). These plugs attenuate the 60-70 hz noise from propeller and exhaust extremely well, better than the isolates did I would say. Above 200 Hz, they seem to be about the same as foam plugs, but the under 100 Hz noise is most of the sound in the cockpit.
I wear them under my Dave Clark ANR conversion headset (a kit from Headsets Incorporated I got to convert my 45 year old Dave Clarks) and can't really tell if the ANR is on or off.
Another tip for improving the performance of earplugs in general is to carry around a tube of lanolin (for baby bottoms) in your flight bag and apply it to the foam. They go in easier, and seal better.
Yes, it's extremely common, both with pilots and with passengers.
As other answers have mentioned, ANC headsets like the Bose A20 are very common with pilots. ANC is especially wonderful for light aircraft where you're sitting a few feet from the engine and often directly behind it. Light aircraft cabins don't provide nearly as much dampening/isolation of engine vibrations as something like an airliner, so the noise levels are quite high and hearing protection is a must. Even those who don't use headsets with active noise cancellation use ones with significant noise-isolating cups that fit tightly around your ears.
Aside from pilots, though, even passengers on commercial flights very frequently use ANC headphones. Many airlines provide ANC headphones to passengers in business and first class on their aircraft that are equipped for long-haul flights. Aside from ones provided by the airline, though, ANC headphones are very common with frequent flyers. The Bose and Sony ones are particularly common. I use the Bose QC35s myself when flying commercial, regardless of what cabin I'm in, as I generally find them to be better than the ANC headphones that the airlines provide. They're wonderful at reducing engine noise, even when you're sitting right beside the engine. They also help significantly for cutting down on cabin noise and are much more comfortable to wear for extended periods than headsets that have tight-fitting noise isolation cups around your ears.