Wave Dash in Super Smash Bros Melee

Ever wonder how the SSBM pros wave dash? Here is a step by step guide to wave dashing.

Steps

Go into training mode and pick Luigi; your computer player is not important.
Pick the stage Final Destination.
Get into the middle of the screen and press the X or Y button to jump. Do it again, but immediately after pressing X or Y, hit down and R. You will notice that smoke appears under your feet, this is good. Practice this until you can do it over and over nonstop. If you are having problems executing, drop the game speed to get the rhythm, then work back to full speed.
Do what you did before, but slightly tilt the control stick to an angle pointing in whatever direction you would like to go in. Do this again and notice how your character slides along the ground.
Practice this, gradually tilting the control stick over a little more each time. You should now be able to go across the entire stage like this, if not, practice.

Never give up if you really want to learn how.
Every character has a place in their jump animation where they are semi-crouching before jumping. That is where you dodge into the ground, even though it seems you're not in the air. Using that you can easily pick up the timing for different characters. Tho only exception to this rule is Bowser (Mewtwo does crouch (sorta) but it doesn't look like a crouch so you'll just have to learn what it looks like).
Practice makes perfect. A "perfect" wave dash is being able to tilt the control stick all the way to the left or right, try to see if you can achieve this.
Don't expect this technique to immediately improve your play. It may take weeks, even months, of practice against other human opponents before you see any noticeable improvement in your gameplay.
Try practicing with a friend, maybe even start a competition to see who can nail down the art of wave dashing first.
Samus is the easiest character to perform the wave dash with, although Luigi's is the most noticeable. Try experimenting with other characters.

Take a break from practicing once in a while. If you don't get it at first, try again later. It won't help you if you're frustrated, remember, practice makes perfect!