So, to my surprise, most people were completely wrong. The following image illustrates the possible angles in Melee:
http://ajpanton.se/ssbm/Others/directions.png
The angles themselves apply for (the ones I've tested):
SmashDI, Airdodge, up+B with Fox, Falco, Zelda, Sheik, Pichu, Pikachu and Mewtwo.
The length of the lines represent the possible lengths of SmashDI, and the number of lengths are limited by the resolution of the control stick (which is actually too low to even allow all possible angles for the shortest distances).
The large areas without possible angles are rounded to the nearest right angle to prevent accidental aims slightly "off-axis" (and unfortunately make it harder to get that perfect 17-degree angle).
My measurements aren't 100% perfect because of the control stick resolution, but they are good enough to see that the angles aren't evenly spaced (so this is not the result of the poor resolution).
Quick facts:
- There are a total of 352 possible angles.
- Within about ±17° from every right angle, the closest right angle will be used instead.
- In other areas the angles are spaced on average 0,65° from each other.
- The angle-differences fluctuate pretty evenly between 0,3° and 1,0° and are located randomly as far as I can see.
- The same pattern is repeated (and mirrored) for every 45-degree sector.
- SmashDI can be shortened down to 70% of its full length, with the number of intermediate lengths only limited by the control stick.