In this paper we prove a result on the structure of the elements of an additive {\it maximum rank distance (MRD) code} over the field of order two, namely that in some cases such codes must contain a semifield spread set. We use this result to classify additive MRD codes in $M_n(\mathbb{F}_2)$ with minimum distance $n-1$ for $n\leq 6$. Furthermore we present a computational classification of additive MRD codes in $M_4(\mathbb{F}_3)$. The computational evidence indicates that MRD codes of minimum distance $n-1$ are much more rare than MRD codes of minimum distance $n$, i.e. semifield spread sets. In all considered cases, each equivalence class has a known algebraic construction.