The harmonic minor is basically thought up to change the natural minor's perfect cadence to be stronger harmonically. The sharpened 7th from the natural minor scale changes the v (minor 7th) in the minor scale to a V7 (dominant 7th) in a harmonic minor scale. So now the perfect cadence becomes V7-i, much stronger than what a natural minor would produce. The V7-i cadence is favored more than the v-i, especially in vocal music like led zeppelin said.

The harmonic minor is sometimes used to harmonize minor chords, especially m/maj7 chords or metal solos. A good note to use in this scale is it's leading tone (or raised 7th). Especially at the end when the cadence above is played. Try playing over the chord progression above and just before you play the 'i' chord play the leading tone then play the root note just as the 'i' chord is played. A very powerful ending to a minor phrase.

The melodic minor is used more in jazz music (traditionally), especially it's modes. Sometimes you'll hear the melodic minor scale referred to as the "Jazz Melodic Minor". An interesting thing about the melodic minor is the scale is different when it is played ascending than descending. Ascending it's the same as a natural minor scale, except you raise the 6th and 7th a half-step. Descending it changes back to a natural minor scale.