Locrian
The diminished mode — unstable and extremely rare
Locrian is the 7th major mode, built on the diminished degree of the major scale. It is the only mode whose root forms a diminished triad (1, ♭3, ♭5) rather than a major or minor one, which makes it inherently unstable. The ♭5 undermines any sense of a tonal center, so phrases rarely resolve.
NoSheet does not practice Locrian. It is extremely rare as a home key — almost no songs are "in Locrian" — and its diminished root triad makes it a poor fit for an ear-training curriculum built on stable major and minor triads. It is included here for completeness.
It appears occasionally in metal or jazz as a passing color, but almost always over a single chord rather than as the key itself.