Much research exists in the continuous setting concerning certain candidate solutions of optimal control problems called singular extremalsSingular extremals, but few results exist in the discrete setting. Existence, uniqueness, and optimality results for singular extremalsSingular extremals of discrete linear control systems with quadratic cost functions were produced in 1971, but very little work has been done since. Our work aims to characterize singular extremalsSingular extremals for a more general class of discrete systems, namely discrete control-affine systems with general cost functions, which arise frequently in practical settings. We seek this characterization with the end goal of applying the theory to the mathematical modeling of spreading phenomena with cellular automataCellular automata, discrete computational models with a notable level of flexibility. We apply the presented theory to epidemiological modeling and wildfire propagation modeling and produce simulations to illustrate our results.