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. 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?

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