I found very different pin guides all over the web, so I decided to go to the source, and looked at the files in the "shop" page for the Micro Arduino Micro — Arduino Official Store
The pinout guide shows PWM on 3, 5, 6, 9, 10, 11, 13 - which matches the spec.
The current software shows no PWM support on pin 4 or 12.
The datasheet doesn't show any PWM (OCx) capability on D4, and D12 is apparently just a complemented version of what's on D6... D3 should have PWM (software and datasheet.)