Let $R$ be a local or graded ring. (If it helps, can assume the ring is "good", e.g. $R=k[[x_1,..,x_p]]$, where $k$ is a field of zero characteristic.)

Let $M$ be a finitely generated $R$-module of finite length, i.e. it is supported only at the origin. How the ``generic" such module looks like?

More precisely, consider the modules of finite length, with presentation matrix of size $m\times n$, i.e. the minimal resolution begins as: $\cdots\rightarrow R^n\rightarrow R^m$. (Thus obviously $dim(R)\le n-m+1$ and the length of the resolution is $dim(R)$.) What are the "generic/typical" values of the invariants? What is the typical betti table? Castelnuovo-Mumford regularity?

(I guess the presentation matrix for the generic such module can be assumed as the matrix of linear forms. Much is known about such matrices for $dim(R)> n-m+1$, but here we have the opposite case.)

A related question: given a morphism of modules $R^n\stackrel{\phi}{\rightarrow} R^m$, what is the analogue of the Eagon-Northcott complex for the annihilator-of-cokernel ideal $ann.coker(\phi)$. (As $dim(R)\le n-m+1$, the ideal $ann.coker(\phi)$ is usually much larger than the Fitting ideal $F_0(\phi)$.)

What is known about the "non-generic" modules? (Some stratification by the ``degeneracy type"?)