The $A_\infty$ and $D_\infty$ plane curve singularities have defining equations $x^2=0$ and $x^2y=0$. These equations are "clearly" natural limiting cases of the equations for $A_n$ singularities $x^2 + y^{n+1}=0$ and $D_n$ singularities $x^2y+y^{n-1}=0$ as $n \to \infty$, since large powers are small in the adic topology. So we're tempted to say that $A_\infty$ and $D_\infty$ are "limits" of the "series of singularities" $\{A_n\}$ and $\{D_n\}$. This was already observed by Arnol'd in 1981, who wrote "Although the series undoubtedly exist, it is not at all clear what a series of singularities *is*."

Have there been any attempts since Arnol'd to make sense out of the phrases in quotes in the previous paragraph? That is:

Are there precise definitions of a "series of singularities", and of the "limit" of a series of singularities, under which $\lim_{n\to \infty} A_n = A_\infty$ and $\lim_{n\to \infty} D_n = D_\infty$?

If the answer is Yes, here's another desideratum: does the notion of "limit" extend to modules/sheaves over the singularities? My motivation here is that the $A_n$ and $D_n$ are (almost) precisely the equicharacteristic hypersurfaces with finite Cohen-Macaulay type (i.e. only finitely many indecomposable MCM modules), while $A_\infty$ and $D_\infty$ are precisely the ones with countable or bounded CM type. I'd really like some statement that each MCM module over the "limit" "comes from" a module "at some finite stage".