  We prove the existence of singular harmonic ${\bf Z}_2$ spinors on
$3$-manifolds with $b_1 > 1$. The proof relies on a wall-crossing formula for
solutions to the Seiberg-Witten equation with two spinors. The existence of
singular harmonic ${\bf Z}_2$ spinors and the shape of our wall-crossing
formula shed new light on recent observations made by Joyce regarding Donaldson
and Segal's proposal for counting $G_2$-instantons.
