A somewhat confusingly named technique, jump cancel shines have little to do with the similarly named Jump cancel up-smash and Jump cancel grab. To do a jump cancel shine, input shine on the very first airborne frames of your jump.
If your input was frame-perfect, you'll end up grounded two frames later than usual; from that position, you can Waveshine as usual. If you jump within those two frames or your shine began on the second airborne frame, you can either double-jump out normally or Triangle jump.
What is the point of this technique, then? Using it, you can shine any time you could jump: most importantly from dash, in shield, or in shine itself.