Let me first recall some definitions from the very first pages of Bourbaki, Commutative Algebra, Chapter 7, "Divisors".

Let $A$ be a (commutative) domain, $K$ its field of fractions. A fractional ideal of $A$ is a finitely generated $A$-submodule of $K$. The set of all non-zero fractional ideals of $A$ is called $I(A)$. On $I(A)$ there is a natural equivalence $\sim$ relation: for two fractional ideals $\mathfrak a$ and $\mathfrak b$ we write ${\mathfrak a} \sim {\mathfrak b}$ if every principal fractional ideal containing $\mathfrak a$ also contains $\mathfrak b$ and vice-versa. The set of equivalence classes in $I(A)$ for this equivalence relation $\sim$ is called $D(A)$; its elements are called divisors of $A$. The multiplication of fractional ideals induces a multiplication on $D(A)$, which makes it a monoid. So $D(A)$ isthe divisor monoid of $A$.

Now on $D(A)$ we define a second equivalence relation, where two elements $d$ and $d'$ of $D(A)$ are equivalent if for some (or equivalently any) representative $\mathfrak a$ of $d$ and for some (or equivalently any) representative $\mathfrak a$ of $d$, one has $$\mathfrak a =\mathfrak a' x \text{ for some }x \in K^\ast.$$ The quotient of $D(A)$ by this equivalence relation clearly inherits the monoid structure of $D(A)$ and is called the divisor class monoid of $A$. Bourbaki doesn't introduce a special notation for it but let us denote it by $DC(A)$.

Bourbaki proves that $D(A)$ is a group (hence also $DC(A)$) if and only if $A$ is totally integrally closed (Theorem 1 of chapter 7). But I am interested in the cases where $A$ is not integrally closed, especially to the cases where $A$ is a noetherian complete domain of Krull dimension 1, or even more especially to the case where $A$ is the completed local ring at a singular point of an algebraic curve over $\mathbb C$. My question is:

Has there been any systematic attempt to compute the divisor class monoid $DC(A)$ for $A$ the completed local ring at a singular point of an algebraic curve? Or at least some example of non trivial computations of such $DC(A)$?

It seems to me that $DC(A)$ is a very natural invariant of a singularity of an algebraic curve. People working in the theory of singularities of algebraic or analytic curves (a vast subject) have certainly met this invariant, but I can't find any reference in the literature. Any pointers, or any suggestion to attack the problem is very welcome.

Remark: I know how to compute $DC(A)$ in simple special cases, for example the case where $A$ is the complete local ring of a cusp, i.e $A=\{f \in \mathbb C[[T]], f'(0)=0\}$. This is Exercise 1 in the exercises of chapter 7, \S1 of Bourbaki. In this case $DC(A)$ is the monoid $\{1,x\}$, where $x$ satisfies $x^2=x$. (Here $x$ can be the class of the ideal $(T,T^2)$ of $A$, for instance). But I'd like to know the answer for more general situations.

Let me first recall some definitions from the very first pages of Bourbaki, Commutative Algebra, Chapter 7, "Divisors".

Let $A$ be a (commutative) domain, $K$ its field of fractions. A fractional ideal of $A$ is a finitely generated $A$-submodule of $K$. The set of all non-zero fractional ideals of $A$ is called $I(A)$. On $I(A)$ there is a natural equivalence $\sim$ relation: for two fractional ideals $\mathfrak a$ and $\mathfrak b$ we write ${\mathfrak a} \sim {\mathfrak b}$ if every principal fractional ideal containing $\mathfrak a$ also contains $\mathfrak b$ and vice-versa. The set of equivalence classes in $I(A)$ for this equivalence relation $\sim$ is called $D(A)$; its elements are called divisors of $A$. The multiplication of fractional ideals induces a multiplication on $D(A)$, which makes it a monoid. So $D(A)$ isthe divisor monoid of $A$.

Now on $D(A)$ we define a second equivalence relation, where two elements $d$ and $d'$ of $D(A)$ are equivalent if for some (or equivalently any) representative $\mathfrak a$ of $d$ and for some (or equivalently any) representative $\mathfrak a$ of $d$, one has $$\mathfrak a =\mathfrak a' x \text{ for some }x \in K^\ast.$$ The quotient of $D(A)$ by this equivalence relation clearly inherits the monoid structure of $D(A)$ and is called the divisor class monoid of $A$. Bourbaki doesn't introduce a special notation for it but let us denote it by $DC(A)$.

Bourbaki proves that $D(A)$ is a group (hence also $DC(A)$) if and only if $A$ is totally integrally closed (Theorem 1 of chapter 7). But I am interested in the cases where $A$ is not integrally closed, especially to the cases where $A$ is a noetherian complete domain of Krull dimension 1, or even more especially to the case where $A$ is the completed local ring at a singular point of an algebraic curve over $\mathbb C$. My question is:

Has there been any systematic attempt to compute the divisor class monoid $DC(A)$ for $A$ the completed local ring at a singular point of an algebraic curve? Or at least some example of non trivial computations of such $DC(A)$?

It seems to me that $DC(A)$ is a very natural invariant of a singularity of an algebraic curve. People working in the theory of singularities of algebraic or analytic curves (a vast subject) have certainly met this invariant, but I can't find any reference in the literature. Any pointers, or any suggestion to attack the problem is very welcome.

Remark: I know how to compute $DC(A)$ in simple special cases, for example the case where $A$ is the complete local ring of a cusp, i.e $A=\{f \in \mathbb C[[T]], f'(0)=0\}$. This is Exercise 1 in the exercises of chapter 7, \S1 of Bourbaki. In this case $DC(A)$ is the monoid $\{1,x\}$, where $x$ satisfies $x^2=x$. (Here $x$ can be the class of the ideal $(T^2,T^3)$ of $A$, for instance). But I'd like to know the answer for more general situations.