  We investigate $N$-extended supersymmetry in one-dimensional quantum
mechanics on a circle with point singularities. For any integer $n$, $N=2n+1$
supercharges are explicitly constructed in terms of discrete transformations,
and a class of singularities compatible with supersymmetry is clarified. In our
formulation, the supersymmetry can be reduced to $M$-extended supersymmetry for
any integer $M<N$. The degeneracy of the spectrum and spontaneous supersymmetry
breaking are also studied.
