I had a lot of noise in every recording made with my Zoom H4n.
Being an audio novice, I was not sure if that was normal for this price range.
After all, I could only notice the noise using good headphones but not with the usual laptop-speakers.
With lots of testing I finally found something by accident.
I notices that the compressor/limiter was set to "Compressor (General)".
A setting I probably did in the very first days.
When I tried to understand all it's features.
I switched the compressor to "off".
Instantly all noise was gone!
Internal microphone, external directional camera microphone and an extremely cheap external lavalier all where crystal clear.
I played around some more and now that the noise was gone I noticed that there was a clearly noticeable 50Hz humming when using the AC-adapter instead of running the H4n on batteries.
Up to now I always had camera and H4n on AC when recording at home because there was no reason to use batteries in my own home.
Update: It seems that the H4n power-supply is not grounded and this causes a 50z humming. A generic power-supply that is grounded is reported to solve this issue.
I hope this finding may help some of the other Zoom H4n users out there with similar noisy recordings.