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We extend Feichtinger's minimality property on smallest non-trivialtime-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 $(0,\infty ]$.

Schatten properties, nuclearity and minimality of shift invariant spaces