How to tag Jazz Clubs and other small place where live music is performmed?

I agree with someoneElse. Back in 2013, I didn't find a suitable tag for describing places like you describe, so I proposed the amenity=music_venue and its response on the tagging mailing was mixed. In the absence of a suitable tag, I continue to use the tag in place where it fits - venues where the primary activity there is live music performances.

Locally, I've tended to go for "amenity=music_venue" for "small music places". What works for you might be different; where I am there are fewer "jazz only" or "blues only" places than there are in e.g. the US or the Netherlands, so a more general tag is more appropriate here. I'd suggest having a look at other places near you and see what tags people have used. In London Ronnie Scott's is a bar; asking Nominatim for a selection of things with Jazz club in the name finds plenty of restaurants, bars and pubs (though of course Nominatim doesn't index everything). Other tags such as "amenity=nightclub" or "amenity=arts_centre" might work for you too depending on the actual nature of the place.