Final Rush Stage.
After passing the sixth checkpoint, grind down the rail in front of it and jump off it to a small platform that has a rocket once you near it.
Ride the rocket which will take you to a platform with an item bubble and go to the nearest platform with a spring.
Jump off that spring and follow the path of platforms until you reach a vertical rail that leads to a higher platform.
The Mystic Melody is located at the end of this platform near one of the shrines that it can be used on.

Wild Canyon Stage.
Go up the air tunnel that takes you outside.
Go to the lonely statue area (The robot Chao will tell you when you are there).
Climb to the top of the statues head and just keep climbing after until you see a painting.
Dig in the middle of the painting and you will go into this secret room and Mystic Melody will be there in plain sight.

Hidden Base Stage.
It is required that Tails has the Booster and Bazooka.
After the first checkpoint, you will encounter a group of little monkey robots. After destroying them, you can either take the left path (which is concealed by a block that can be destroyed) or the right path. Either way, they will both lead to the same path.
At the end of this path some Iron Boxes are found in the wall. Destroy them and the Mystic Melody wil be found behind.

Final Chase Stage.
After passing the third checkpoint and entering the segment with the green cylinders, travel up the cylinders to the highest point possible on them in that area.
At the top of one of them, you'll see a cylinder with an electrical fence go to the top of this cylinder. At the top, there will be a small platform with a small shrine and the Mystic Melody.

Dry Lagoon Stage.
In order to obtain it, Rouge has to have the Pick Nails.
At the start of the level, glide to the right (behind the GUN robots with the turtle).
Dig into the middle of the painting on the wall to find a small room with the Mystic Melody.

Sand Ocean Stage.
Dr. Eggman has to have the Jet Engine.
At the beginning of the level, stand on the turning platform and wait for it to have the long edge of itself to the left. Hover over to the small platform that is to the left, you will find the Mystic Melody there.