Locrian #2 (also called Aeolian b5) is the preferred scale for half-diminished chords in jazz. Compared to regular Locrian, the natural 2nd makes it much more usable - less harsh, more elegant.
Scale Formula
1 2 b3 4 b5 b6 b7 - Locrian with natural 2. Semitone pattern: W H W H W W W
Fretboard (C Locrian #2)
Toggle to interval view and compare with regular Locrian. The natural 2 (instead of b2) removes the harshest dissonance, making this mode far more musical in practice.
Common Chords
Half-diminished 7th (1 b3 b5 b7), Minor 7b5 (1 b3 b5 b7), Minor 9 b5.
Diatonic Harmony
Improvisation Context
Play over half-diminished (m7b5) chords. This is the jazz default for half-diminished - preferred over standard Locrian because the natural 2 avoids the dissonant b9 interval against the root.
Practice
Play C Locrian #2 over a Cm7b5 chord and compare it with regular C Locrian. The natural 2 (D vs Db) makes a dramatic difference in usability. Then apply it in context: over a Dm7b5-G7-Cm progression (ii-V-i in C minor), play D Locrian #2 over the Dm7b5. The natural 2 (E) is a chord tone of the following G7, creating smooth voice leading into the V chord.