  A graph is {\em locally irregular} if no two adjacent vertices have the same
degree. A {\em locally irregular edge-coloring} of a graph $G$ is such an
(improper) edge-coloring that the edges of any fixed color induce a locally
irregular graph. Among the graphs admitting a locally irregular edge-coloring,
i.e., {\em decomposable graphs}, only one is known to require $4$ colors, while
for all the others it is believed that $3$ colors suffice. In this paper, we
prove that decomposable claw-free graphs with maximum degree $3$, all cycle
permutation graphs, and all generalized Petersen graphs admit a locally
irregular edge-coloring with at most $3$ colors. We also discuss when $2$
colors suffice for a locally irregular edge-coloring of cubic graphs and
present an infinite family of cubic graphs of girth $4$ which require $3$
colors.
