Classifying complete C-subalgebras of C[[t]]

We address the problem of classifying complete C-subalgebras of C[[t]].A discrete invariant for this classification problem is the semigroup of orders of the elements in a given C-subalgebra.Hence we can define the space R-Gamma of all C-subalgebras of 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 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 R-Gamma is always an affine space, and for general Gamma we describe the stratification of R-Gamma by embedding dimension.We also describe the natural map from R(Gamma)to the Zariski moduli space in some special cases. Explicit examples are provided throughout.