  The arithmetic degree, the smallest extended degree, and the homological
degree are invariants that have been proposed as alternatives of the degree of
a module if this module is not Cohen-Macaulay. We compare these degree
functions and study their behavior when passing to the generic initial or the
lexicographic submodule. This leads to various bounds and to counterexamples to
a conjecture of Gunston and Vasconcelos, respectively. Particular attention is
given to the class of sequentially Cohen-Macaulay modules. The results in this
case lead to an algorithm that computes the smallest extended degree.
