  We show new local $L^p$-smoothing estimates for the Schr\"odinger equation
with initial data in modulation spaces via decoupling inequalities.
Furthermore, we probe necessary conditions by Knapp-type examples for
space-time estimates of solutions with initial data in modulation and
$L^p$-spaces. The examples show sharpness of the smoothing estimates up to the
endpoint regularity in a certain range. Moreover, the examples rule out global
Strichartz estimates for initial data in $L^p(\mathbb{R}^d)$ for $d \geq 1$ and
$p > 2$, which was previously known for $d \geq 2$. The estimates are applied
to show new local and global well-posedness results for the cubic nonlinear
Schr\"odinger equation on the line. Lastly, we show $\ell^2$-decoupling
inequalities for variable-coefficient versions of elliptic and non-elliptic
Schr\"odinger phase functions.
