Given a moduli space $M$ of some smooth algebraic geometric object such as curves, surfaces, etc. Let $\overline{M}$ be a compactification of $M$. Then, $\overline{M}\setminus M$ introduces singular objects in our moduli. The question is: What is the use of knowing the singularities parametrized by the boundary of the compactified moduli space i.e $\overline{M} \setminus M$. Usually $M$ has diferent compactifications, and so different " limit singular objects". Does this difference mean something?

For example: the smooth genus $g=3$ have the $\overline{M_3}$ compactification with only stable curves in the boundary, but is possible to find another compactification by the GIT analysis of degree four plane curves. The singular curves present in the boundaries are quite different. What is the use of having an explicit descriptions of them?