  Given a graph $G$, an injective edge-coloring of $G$ is a function $\psi:E(G)
\rightarrow \mathbb N$ such that if $\psi(e) = \psi(e')$, then no third edge
joins an endpoint of $e$ and an endpoint of $e'$. The injective chromatic index
of a graph $G$, written $\chi_{inj}'(G)$, is the minimum number of colors
needed for an injective edge coloring of $G$. In this paper, we investigate the
injective chromatic index of certain classes of degenerate graphs. First, we
show that if $G$ is a $d$-degenerate graph of maximum degree $\Delta$, then
$\chi_{inj}'(G) = O(d^3 \log \Delta)$. Next, we show that if $G$ is a graph of
Euler genus $g$, then $\chi_{inj}'(G) \leq (3+o(1))g$, which is tight when $G$
is a clique. Finally, we show that the oriented chromatic number of a graph is
at most exponential in its injective chromatic index. Using this fact, we prove
that the oriented chromatic number of a graph embedded on a surface of Euler
genus $g$ has oriented chromatic number at most $O(g^{6400})$, improving the
previously known upper bound of $2^{O(g^{\frac{1}{2} + \epsilon})}$ and
resolving a conjecture of Aravind and Subramanian.
