Melodic or Harmonic Minor

Q: I'm enjoying your Infinite Guitar book very much lately but I'm wondering why you dedicate so much time to Melodic Minor and not so much to Harmonic Minor. Why is Melodic Minor more important?

A: Good question. There isn't anything wrong with Harmonic Minor so to speak but Melodic Minor has more applications. Only one mode of Harmonic Minor is exclusively used and that mode is the 5th one, sometimes named the Phrygian Dominant mode. As an example, E to E in the A Harmonic Minor scale: E-F-G#-A-B-C-D. You would play it over an E7 or E7b9 chord and more often than not, it would get used over a V chord in a minor progression. For example in a rock application: Amin-G-F-E, you would want to play A natural minor over the first three chords and A Harmonic Minor over the final E chord. In a Jazz application, you would play it over a iio-V7 in A minor, as in Bmin7b5-E7b9-Amin7 (over the last chord, you wouldn't generally play A harmonic minor but A natural minor or A dorian. You can use Harmonic Minor over a i chord (as in A Harmonic Minor over an Amin chord) but it sounds like you are playing Mexican or Middle Eastern music.
I might do this if I need to play something blatantly stinky, like when a director asks me to play something that sounds like a snake charmer.

On to Melodic Minor (click on the links for more information)

Dorian b2: B-C-D-E-F#-G#-A (1-b2-b3-4-5-6-b7): This mode gets used over a 13b9sus4 chord or a b9sus4 chord. It's not a very common chord but has a great sound.

Lydian Augmented: C-D-E-F#-G#-A-B (1-2-3-#4-#5-6-7): This gets used over a Cmaj7#5 chord. You can play it like a E/C slash chord. 

Lydian Dominant: D-E-F#-G#-A-B-C (1-2-3-#4-5-6-b7): Play this over a D7, D9, D13 chord but especially over a D7#11 chord. You can mix this scale up with a Blues scale over a vamp for some interesting contrast.

Locrian #2: F#-G#-A-B-C-D-E (1-2-b3-4-b5-b6-b7): This is a much better choice over a min7b5 chord than locrian. 

Altered: G#-A-B-C-D-E-F# (1-b2-#2-3-b5-#5-b7): Perfect for any altered dominant chord, as in 7(#5,#9), 7(b5,b9), 7(#5,b9), 7(b5,#9)

Most common are the Lydian Dominant and Altered mode but the other modes can be used quite effectively as well. For example, over a minor iio-V in A minor, you could play a B locrian #2 mode (D Melodic Minor) over the Bmin7b5, and the E altered mode (F Melodic Minor) over the E7b9 chord. You could even get away with playing A Melodic Minor over the i chord if you do it right (this works because A Melodic Minor looks a lot like A dorian, actually a dorian scale with a major 7th).