  We address the problem of classifying complete $\mathbb{C}$-subalgebras of
$\mathbb{C}[[t]]$. A discrete invariant for this classification problem is the
semigroup of orders of the elements in a given $\mathbb{C}$-subalgebra. Hence
we can define the space $\mathcal{R}_{\Gamma}$ of all $\mathbb{C}$-subalgebras
of $\mathbb{C}[[t]]$ with semigroup $\Gamma$. After relating this space to the
Zariski moduli space of curve singularities and to a moduli space of global
singular curves, we prove that $\mathcal{R}_{\Gamma}$ is an affine variety by
describing its defining equations in an ambient affine space in terms of an
explicit algorithm. Moreover, we identify certain types of semigroups $\Gamma$
for which $\mathcal{R}_{\Gamma}$ is always an affine space, and for general
$\Gamma$ we describe the stratification of $\mathcal{R}_{\Gamma}$ by embedding
dimension. We also describe the natural map from $\mathcal{R}_{\Gamma}$ to the
Zariski moduli space in some special cases. Explicit examples are provided
throughout.
