  We investigate Sobolev spaces $W^{1,\Phi}$ associated to Musielak-Orlicz
spaces $L^\Phi$. We first present conditions for the boundedness of the Voltera
operator in $L^\Phi$. Employing this, we provide necessary and sufficient
conditions for $W^{1,\Phi}$ to contain isomorphic subspaces to $\ell^\infty$ or
$\ell^1$. Further we give necessary and sufficient conditions in terms of the
function $\Phi$ or its complementary function $\Phi^*$ for reflexivity, uniform
convexity, $B$-convexity and superreflexivity of $W^{1,\Phi}$. As corollaries
we obtain the corresponding results for Orlicz-Sobolev spaces $W^{1,\varphi}$
where $\varphi$ is an Orlicz function, the variable exponent Sobolev spaces
$W^{1,p(\cdot)}$ and the Sobolev spaces associated to double phase functionals.
