  A star edge coloring of a graph is a proper edge coloring with no $2$-colored
path or cycle of length four. The star chromatic index $\chi'_{st}(G)$ of $G$
is the minimum number $t$ for which $G$ has a star edge coloring with $t$
colors. We prove upper bounds for the star chromatic index of complete
bipartite graphs; in particular we obtain tight upper bounds for the case when
one part has size at most $3$. We also consider bipartite graphs $G$ where all
vertices in one part have maximum degree $2$ and all vertices in the other part
has maximum degree $b$. Let $k$ be an integer ($k\geq 1$), we prove that if
$b=2k+1$ then $\chi'_{st}(G) \leq 3k+2$; and if $b=2k$, then $\chi'_{st}(G)
\leq 3k$; both upper bounds are sharp.
  Finally, we consider the well-known conjecture that subcubic graphs have star
chromatic index at most $6$; in particular we settle this conjecture for cubic
Halin graphs.
