We investigate when a maximum distance separable ($MDS$) code over $F_q$ is also completely regular ($CR$). For lengths $n=q+1$ and $n=q+2$ we provide a complete classification of the $MDS$ codes that are $CR$ or at least uniformly packed in the wide sense ($UPWS$). For the more restricted case $n\leq q$ with $q\leq 5$ we obtain a full classification (up to equivalence) of all nontrivial $MDS$ codes: there are none for $q=2$; only the ternary Hamming code for $q=3$; four nontrivial families for $q=4$; and exactly six linear $MDS$ codes for $q=5$ (three of which are $CR$ and one admits a self-dual version). Additionally, we close two gaps left open in a previous classification of self-dual $CR$ codes with covering radius $ρ\leq 3$: we precisely determine over which finite fields the $MDS$ self-dual completely regular codes with parameters $[2,1,2]_q$ and $[4,2,3]_q$ exist.