  The star chromatic index of a multigraph $G$, denoted $\chi'_{s}(G)$, is the
minimum number of colors needed to properly color the edges of $G$ such that no
path or cycle of length four is bi-colored. A multigraph $G$ is star
$k$-edge-colorable if $\chi'_{s}(G)\le k$. Dvo\v{r}\'ak, Mohar and \v{S}\'amal
[Star chromatic index, J. Graph Theory 72 (2013), 313--326] proved that every
subcubic multigraph is star $7$-edge-colorable. They conjectured in the same
paper that every subcubic multigraph should be star $6$-edge-colorable. In this
paper, we first prove that it is NP-complete to determine whether
$\chi'_s(G)\le3$ for an arbitrary graph $G$. This answers a question of Mohar.
We then establish some structure results on subcubic multigraphs $G$ with
$\delta(G)\le2$ such that $\chi'_s(G)>k$ but $\chi'_s(G-v)\le k$ for any $v\in
V(G)$, where $k\in\{5,6\}$. We finally apply the structure results, along with
a simple discharging method, to prove that every subcubic multigraph $G$ is
star $6$-edge-colorable if $mad(G)<5/2$, and star $5$-edge-colorable if
$mad(G)<24/11$, respectively, where $mad(G)$ is the maximum average degree of a
multigraph $G$. This partially confirms the conjecture of Dvo\v{r}\'ak, Mohar
and \v{S}\'amal.
