The harmonic minor scale has just one tonally effective mode and that is the scale conventionally known as the harmonic minor scale. It is spelled, in numerical form relative to the major scale.
1 2 flat3 4 5 flat6 7 Notes
i iidim flatIII+ iv V flatVI viidim Chords
If the tonic is c, the notes and chords are:
c d eflat f g aflat b
c d dim Eflat+ f G Aflat b
The harmonic minor scale is well known to common practice classical music because it is the harmonic foundation of minor mode music. It is, however, avoided as the melodic foundation because of the "unmelodic" augmented second found between its sixth and seventh degrees.
Conventionally when the sixth degree proceeds to the seventh the sixth degree is raised by a chromatic semitone, and when the seventh degree proceeds to the sixth the seventh degree is lowered by a chromatic semitone. Both of these devices transform the augmented second into a major second.
These devices are used simply to smooth the melodic line without disturbing tonal function in the scale too drastically, although repeated use of the natural sixth will weaken the tonal function of the scale. It is not necessary though, and the melodic leap between the sixth and seventh degree can be used as an interesting feature.
The reason that the harmonic minor scale is used as the harmonic foundation of the minor mode is that, despite its melodic deficiencies, its tonality is very powerful and unambiguous, whereas the tonality of the aeolian mode is weak and easily displaced, and the tonality of the melodic minor is even weaker and more ambiguous. In a sense the harmonic minor scale is the "default" scale to which the melodic variations must return in order for the tonality to be maintained. By using it as the harmonic resource for the minor mode one is emphasizing its fundamental role in maintaining tonal function.
There are, of course seven modes of the harmonic minor scale just as there are with the diatonic scale, none of them have common names, and it is only the harmonic minor which is tonally effective.