Defined, in this case, as "museums you probably don't know about/won't come across on most museum lists/probably shouldn't visit but eventually will." Educate yourself!
The Museum of Oriental Musical Instruments
If you're into Asian musical instruments but not into going to Minhang, there's also this small museum founded by the Shanghai Conservatory of Music, located in a little garden villa off Gao'an Lu. They've collected about 500 instruments and curated them into four exhibition areas; ancient Chinese instruments, Chinese minority folk instruments, modern Chinese instruments, and foreign national instruments. The displays include everything from a Jiahu bone flute to replica Tang Dynasty instruments to an Indonesian gamalan, as well as instruments from South Korea, Japan, Thailand, and India. It's a little hard to find: when you arrive at the start of Lane 18 on Gao'An Lu, go down until you reach the large black gates of the garden villa on your left.
The Museum of Oriental Musical Instruments
No. 2, Lane 18 Gao'an Lu, near...