Ok, let me try to say something intelligent about the 1-dimensional case.

As has already been noted, the normal Noetherian 1-dimensional rings are finite products of Dedekind domains. So what about the non-normal case (say for reduced rings)?

## Pass to the normalization (dimension = 1)

Suppose that $R$ is an excellent purely 1-dimensional reduced ring. Let $R^{\mathrm{N}}$ be the normalization. So we know that $R^{\mathrm{N}}$ is a finite product of Dedekind domains. How do we reconstruct $R$ from it?

It follows that $R$ is always the pullback of a diagram $\{ R^{\mathrm{N}} \to R^{\mathrm{N}}/I \leftarrow S \}$ where $R^{\mathrm{N}}/I$ and $S$ are zero-dimensional (which say we've already classified) and the map $S \to R^{\mathrm{N}}/I$ is injective. To see this, see my answer here and my answer here.

On the other hand, if $A$ is Noetherian normal and purely 1-dimensional, $A/I$ and $B$ are zero dimensional, and $B \to A/I$ is injective, then $\{ A \to A/I \leftarrow B \}$ gives you a 1-dimensional reduced Noetherian ring. So this is at some level a classification (in the second answer, a canonical choice of the $A/I$ and $B$ is given).

On the other hand, if the map $B \to A/I$ is *not* injective (but still a map between zero dimensional rings), then this should actually classify 1-dimensional rings that are generically reduced.

## Two dimension and higher

**Edit:** I originally said this was hopeless, but maybe that's too pessimistic. There are approaches at some level. Here are some vague thoughts and directions to look up for references.

Ok, so probably it be reasonable to localize (and maybe also complete). Then what do you do? There are some perhaps even reasonable results. It might be of course you have to leave the world of *algebraic structure* and pass to *geometric structure*.

Resolution of singularities (if it is known to exist) says something, but doesn't tell you everything you might want (certainly the structure of the exceptional divisor, which is heavily studied, is less information than isomorphism after completion).

One can talk about the equisingularity theory as introduced by Zariski (which tries to break up the variety into strata where all the singularities are the same). This might be really relevant to you and there are plenty of recent papers on it in the commutative algebra. This is still a really hard area in my opinion.

I remember Hans Schoutens talking about a version of this problem in an AMS special session once, the most relevant version of the paper I can find is this one: click HERE. I don't quite see how one could use the sort of thing that Schoutens proves but maybe if it was further developed (or maybe you see ways to use it).