A $q$-ary code of length $n$, size $M$, and minimum distance $d$ is called an $(n,M,d)_q$ code. An $(n,q^{k},n-k+1)_q$ code is called a maximum distance separable (MDS) code. In this work, some MDS codes over small alphabets are classified. It is shown that every $(k+d-1,q^k,d)_q$ code with $k\geq 3$, $d \geq 3$, $q \in \{5,7\}$ is equivalent to a linear code with the same parameters. This implies that the $(6,5^4,3)_5$ code and the $(n,7^{n-2},3)_7$ MDS codes for $n\in\{6,7,8\}$ are unique. The classification of one-error-correcting $8$-ary MDS codes is also finished; there are $14$, $8$, $4$, and $4$ equivalence classes of $(n,8^{n-2},3)_8$ codes for $n=6,7,8,9$, respectively. One of the equivalence classes of perfect $(9,8^7,3)_8$ codes corresponds to the Hamming code and the other three are nonlinear codes for which there exists no previously known construction.