  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 $(0,\infty ]$.
