I think that my question is pretty self-explanatory. I‘m interested in both general-aviation and commercial jet pilots.
it depends on the flight. With a low altitude lower speed ( <200kts) GA flights they can still be useful and can serve as a usefull backup. Once you travel faster than 200 kts, and higher than about 12,000 you end up switching cell towers too fast and the phone uses a lot of power trying to get as strong enough signal. If connected to the planes power this will increase the possibility of the phone creating radio interference (analog audio or nav signals)
The item "Mobile phone ... OFF" is (or at least was) actually a part of the Before Engine Start checklist on several commercial aircraft types. I tried to come up with a reference, but most photos found online only refer to PC simulator software. However, there seems to be one actual photo of the checklists on the cockpit table from a real A320 aircraft found here.
Personally I have seen Airbus (A320/A330/A340) pilots of at least one major European carrier going through that checklist item and accordingly checking and verbally confirming that both their mobile phones were switched off.
(Disclaimer: this experience was 10+ years ago. I don't know whether policies have changed since.)
I am a novice, non-professional pilot. I have never seen anyone (instructors, examiners, other pilots) switch their phone to airplane mode, and in fact it is not uncommon to make cell phone calls from the air. It is also extremely common to use data connected applications (e.g. ForeFlight) from the air.
That being said, it seems like a pretty good idea to keep it in airplane mode when you aren't using it, especially when not navigating visually. As mentioned in some other answers, cellphones definitely can cause radio interference.