  We establish sharp bounds for the second moment of symmetric-square
$L$-functions attached to Hecke Maass cusp forms $u_j$ with spectral parameter
$t_j$, where the second moment is a sum over $t_j$ in a short interval. At the
central point $s=1/2$ of the $L$-function, our interval is smaller than
previous known results. More specifically, for $|t_j|$ of size $T$, our
interval is of size $T^{1/5}$, while the previous best was $T^{1/3}$ from work
of Lam. A little higher up on the critical line, our second moment yields a
subconvexity bound for the symmetric-square $L$-function. More specifically, we
get subconvexity at $s=1/2+it$ provided $|t_j|^{6/7+\delta}\le |t| \le
(2-\delta)|t_j|$ for any fixed $\delta>0$. Since $|t|$ can be taken
significantly smaller than $|t_j|$, this may be viewed as an approximation to
the notorious subconvexity problem for the symmetric-square $L$-function in the
spectral aspect at $s=1/2$.
