We study separating matchings in graphs, that is, matchings whose removal increases the number of connected components, and focus on determining the maximum size of such a matching in a graph $G$, denoted by $\mathrm{mms}(G)$. We show that every subcubic graph admits a separating matching, except for exactly eight graphs, which allows us to focus on bounding $\mathrm{mms}(G)$ for cubic graphs. Our main results show that every cubic graph $G$ on $n$ vertices that admits a separating matching satisfies $\mathrm{mms}(G) \ge n/2 - 2$. For bipartite cubic graphs, assuming a conjecture of Funk, the problem reduces to a recursively defined class $\mathcal{F}$, for which we prove that $\mathrm{mms}(G) \ge n/2 - 1$, up to four exceptional graphs. In contrast, we show that every claw-free cubic graph satisfies $\mathrm{mms}(G) = n/2$. These results extend previous work on matching cuts and disconnected $2$-factors, and provide the first systematic study of maximum separating matchings.