Consider the action of $GL_n(R)$ on $M_{n \times n}(R)$ by conjugation, where $R$ is a ring (or field)? How can we classify the orbits? To what extent does the characteristic polynomial and the trace classify the orbits?

For a field, this is given by the rational canonical form (see Section 7.2 of Hoffman and Kunze's Linear Algebra, for example). Even in this case, the trace and characteristic polynomial are quite weak as invariants. What you need are the invariant factors. For general rings, this is very hard and often a wild classifcation problem. For rings like $\mathbf Z/p^k\mathbf Z$ look at this paper. 

