CS70 wrote: Since I don't live in the UK I have no opinion about the laws where you are, but the fellows here have indeed quite a lot of experience with that type of questions, and if they say that it's moot, it's most likely moot.
On the technical side, if the noise is loud enough to annoy you, it's odd that it cannot be captured by a mic, unless it's really, really low frequencies (which you would perceive more as a vibration than a sound).
And almost certainly is not the preamp noise floor of the Zoom which is the problem, and what you are hearing is the ambient noise in you room.
Dynamic mics (like a 58) aren't that great for high frequency noise as they roll off the highs, but once again it's hard to say what you mean by "high frequency". It seems unlikely that the sound that annoys you is entirely over the 13-15KHz threshold. You might fare a bit better with a condenser mic, which covers more than the entire audible spectrum, but again it's odd that you don't get anything usable with a 58. It has a cardioid pattern so it won't pick up much from rear and sides, but if you point it towards the neightbour flat, it should. You might try a more sensitive dynamic microphone (there's some), or one with an omni pattern.
But might it be that it's more of a perceptive issue, like you're trying to concentrate and suddenly there is a relatively faint but audible door slam which annoys you? Or do you hear it really loud? Because if you do, so it will even the cheapest microphone.
You may also to equalize and compress the recording a little bit to reveal more details, but if you start manipulating it too much it won't be really that valid even as indicative evidence..
I'm almost certain it's the Zoom's internal noise floor that's the problem, as there's nothing in my bedroom making any noise that would cause the hiss in my recording and the noise I'm trying to record is really quite loud and intrusive, certainly louder than a lot of ambient nature sounds that people might record, so I don't think the Zoom would be any good for that. The SM58 doesn't pick up the noise at all but I also don't get much, if any, hiss using the external input which makes me think the internal mic uses a different pre-amp. I really need a different external mic that is designed to pick up high frequencies though.
I've still got quite good hearing for high-frequencies, although I've also got tinnitus and auditory processing disorder, the latter of which makes it difficult to understand speech when there's a lot of background noise like in a busy pub.
Actually I've just used a frequency generator on my PC to compare with my recording and it seems that the noise I'm trying to record isn't high-frequency at all, it's around 400hz, which makes it even more strange that my SM58 isn't picking it up.