Mystic Melody for Everyone
Sonic: Go to Final Rush. At the 6th check point, you will land on a platform with two rails, both leading forward. (If you were fast, it will take about 3 minutes to get here.) One is purple, the other, orange. Take the ORANGE one and hold the [B] button for the entirety of the grind. Done correctly, this should toss you onto a platform with a rocket on it. Take the rocket, and you will receive your new toy!
Tails: it's in Hidden Base, below the SECOND point marker. When it's in sight, instead of jumping towards it, turn around and go backwards about 50 or 100 'feet', then walk left until you come across a mural. Drop down from here (hover, if you prefer.) You will find yourself on a walkway with iron crates blocking a door. You should know what to do there.
Knuckles: get the Shovel Claws from Pumpkin Hill, then go back to Wild Canyon. Glide into the wind, and turn left, towards the Lonely Statue Area. Climb above the Lonely Statue and dig into the center of the painting. Knuckles will find a secret passage into a hidden room with the powerup.
Shadow: it's in Final Chase. Run to the top of every vertical drum. At the top of one, you will see another verical drum with an electric net around the center. Shadow must jump to that drum ABOVE the electric net to net his prize.
Dr. Eggman: His melody can be learned by hovering LEFT of the first rotating platform in Sand Ocean. Obviously, he'll need to have the Jet Engine to be able to reach it!
Rouge: She needs the Pick Nails to get her Mystic Melody. Take them to Dry Lagoon. Find a painting on the wall nearest the turtle being bullied by GUN Hunters. Dig in!