Classifying complete $\mathbb{C}$-subalgebras of $\mathbb{C}[[t]]$

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 ${\mathcal{R}}_{\mathrm{\Gamma }}$ of all $\mathbb{C}$-subalgebras of $\mathbb{C}[[t]]$ with semigroup $\mathrm{\Gamma }$. After relating this space to the Zariski moduli space of curve singularities and to a moduli space of global singular curves, we prove that ${\mathcal{R}}_{\mathrm{\Gamma }}$ 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 $\mathrm{\Gamma }$ for which ${\mathcal{R}}_{\mathrm{\Gamma }}$ is always an affine space, and for general $\mathrm{\Gamma }$ we describe the stratification of ${\mathcal{R}}_{\mathrm{\Gamma }}$ by embedding dimension. We also describe the natural map from ${\mathcal{R}}_{\mathrm{\Gamma }}$ to the Zariski moduli space in some special cases. Explicit examples are provided throughout.