Schatten properties, nuclearity and minimality of shift invariant spaces

We extend Feichtinger's minimality property on smallest non-trivial time-frequency shift invariant Banach spaces, to the quasi-Banach case. Analogous properties are deduced for certain matrix classes. We use these results to prove that $\operatorname{Op}(a)$ is a Schatten-$q$ operator from $M^\infty$ to $M^p$ and $r$-nuclear operator from $M^\infty$ to $M^r$ when $a\in M^r$ for suitable $p$, $q$ and $r$ in $(0,\infty ]$.