I am white... why do I get the feeling that everyone on the internet doesn't think of me as being white? >.>
Waveshine -> Dash -> Jump -> Shine
Shining in the air actually causes Fox and Falco to immediately move downward slightly. If you have just left the ground from a jump (or are halfway through a platform), shining will move you back onto the ground so that you can jump cancel with your first jump instead of your second. This is the whole basis behind why multishining works. So what I am suggesting is that you dash to move next to the opponent, then you jump, then you shine on the FIRST FRAME after you leave the ground, and it will be a grounded shine. When Fox jumps, he spends 3 frames on the ground in a crouching animation (called jumpsquat). If you shine on frame 4, you will land back on the ground.
Hopefully that clears up what I was talking about. If you want to experiment with this yourself, go pick Falco (he has 5 frames of jumpsquat instead of 3, so it is easier to do) and Kongo Jungle 64. Stand on the rotating platforms, and when the one you are standing on is rising, try to jump and then shine right as you leave the ground. The platforms moving up make it easier to grounded shine because even if you are a frame or two late with your shine, you are still close enough to the platform to land from the shine's small movement.
Here's a little
demo (I'm Falco).
Notice how it looked like I was glued to the platform immediately after the third shine, and that allowed me to JC it into another grounded shine. If I had gone too high or low relative to the platform, I would have simply shined, stayed airborne, and then double jumped out of the shine.