  Let $M$ be a compact oriented 3-dimensional smooth manifold. In this paper,
we construct a moduli space consisting of pairs $(\Sigma, \psi)$ where $\Sigma$
is a $C^1$-embedding simple closed curve in $M$, $\psi$ is a
$\mathbb{Z}/2$-harmonic spinor vanishing only on $\Sigma$, and
$\|\psi\|_{L^2_1}\neq 0$. We prove that when $\Sigma$ is $C^2$, a neighborhood
of $(\Sigma, \psi)$ in the moduli space can be parametrized by the space of
Riemannian metrics on $M$ locally as the kernel of a Fredholm operator.
