I recorded a video in a club, which means that there is a music background which is copyrighted. The club being ~105 dB and the record device being simply my phone, the music quality is pretty bad. How bad does it need to be for me to avoid copyright claim on YouTube?
I doubt this answers
How bad does it need to be for me to avoid copyright claim on YouTube? but maybe explains the downvote.
there is no definitive test for determining whether your use of another's copyrighted work is a fair use. In effect, all copyright infringement, or not, is a matter of opinion, so presumably the OP would be off topic on that count.
If 'opinion' is too imprecise, then maybe 'a legal matter' - for which WA is hardly appropriate, and OP would be too localised anyway.
However, maybe some consolation is this extract:
Capturing copyrighted material incidentally or accidentally
Video makers often record copyrighted sounds and images when they are recording sequences in everyday settings. For instance, they may be filming a wedding dance where copyrighted music is playing, capturing the sight of a child learning to walk with a favorite tune playing in the background, or recording their own thoughts in a bedroom with copyrighted posters on the walls. Such copyrighted material is an audio-visual found object. In order to eliminate this incidentally or accidentally captured material, makers would have to avoid, alter, or falsify reality.
PRINCIPLE: Fair use protects the creative choices of video makers who seek their material in real life. Where a sound or image has been captured incidentally and without pre-arrangement, as part of an unstaged scene, it is permissible to use it, to a reasonable extent, as part of the final version of the video. Otherwise, one of the fundamental purposes of copyright--to encourage new creativity--would be betrayed.
LIMITATION: In order to take advantage of fair use in this context, the video maker should be sure that the particular media content played or displayed was not requested or directed; that the material is integral to the scene or its action; that the use is not so extensive that it calls attention to itself as the primary focus of interest; and that where possible, the material used is properly attributed.
So the music quality seems irrelevant but in my opinion (I am no more than a "barrack-room" lawyer) there is no significant risk of copyright infringement in the scenario you describe.