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In \cite[Section 5, p.32]{Arnold-1998}, Arnold writes: "Classification ofsingularities of curves can be interpreted in dual terms as a description of'co-artin' subalgebras of finite co-dimension in the algebra of formal seriesin a single variable (up to isomorphism of the algebra of formal series)." Inthe paper, such a description is obtained but up to isomorphism of algebraiccurves (i.e. this description is finer).  Let $K$ be an algebraically closed field of arbitrary characteristic. The aimof the paper is to give a classification (up to isomorphism) of the set ofsubalgebras $\mathcal{A}$ of the polynomial algebra $K[x]$ that contains theideal $x^mK[x]$ for some $m\geq 1$. It is proven that the set $\mathcal{A} =\coprod_{m, \Gamma }\mathcal{A} (m, \Gamma )$ is a disjoint union of affinealgebraic varieties (where $\Gamma \coprod \{0, m, m+1, \ldots \}$ is thesemigroup of the singularity and $m-1$ is the Frobenius number). It is proventhat each set $\mathcal{A} (m, \Gamma )$ is an affine algebraic variety andexplicit generators and defining relations are given for the algebra of regularfunctions on $\mathcal{A} (m ,\Gamma )$. An isomorphism criterion is given forthe algebras in $\mathcal{A}$. For each algebra $A\in \mathcal{A} (m, \Gamma)$,explicit sets of generators and defining relations are given and theautomorphism group ${\rm Aut}_K(A)$ is explicitly described. The automorphismgroup of the algebra $A$ is finite iff the algebra $A$ is not isomorphic to amonomial algebra, and in this case $|{\rm Aut}_K(A)|<{\rmdim}_K(A/\mathfrak{c}_A)$ where $\mathfrak{c}_A$ is the conductor of $A$. Theset of orders of the automorphism groups of the algebras in $\mathcal{A} (m ,\Gamma )$ is explicitly described.

The Question of Arnold on classification of co-artin subalgebras in singularity theory