For the Chinese, music is an obsession, especially Peking Opera and the classical sounds of the traditional Chinese instruments like the gu zheng, er hu and others. You'll find music flowing through the city from passing taxis where the drivers sing along to operas that are hundreds of years old, and from the multitude of opera and concert halls as well as jazz bars (a newer but well received for of performance music for Shanghai).
The Shanghai Concert Hall is one of the best places to catch live music showings in the form of opera, classical recitals and performances and even jazz. Likewise, the Chinese Opera School trains the city's up and coming performance stars to hone in on their skills and belt out opera tunes to please the masses.
Smaller theaters have more intimate performances at times, and the Yifu Theater and KunJu Theater are two such examples, though you may need to book early to get a spot.
For jazz, you'll want to start at the Cigar Jazz Wine bar, or the Peace Hotel. The House of Jazz and Blues is another fantastic place to absorb American/French and now Chinese infusions of this musical genre.
For something truly unique, try the Ai Ya Japanese nightlife scene with life music from across the pond.
9 Shaoxing Lu
(Pu Xi District)
370 Hua Shan Road
Jing'An Hotel, San Diego Hall
Cao Xi Bei Lu 1200
Huating Hotel and Towers (Huating Binguan), Xuhui
222 Yan An Road
Bund Center 50/F Office Tower
60 Fuzhou Road
Near the Bund