This work is concerned with the study of the regularity and the statistical properties of Lyapunov exponents of random locally constant linear cocycles. We investigate both the case when the support of the underlying measure consists of only invertible matrices, as well as the case when it also contains non-invertible matrices. It turns out that these two settings exhibit strikingly different behaviors. In the invertible case we study the regularity of the Lyapunov exponent as a function of the underlying measure relative to two different topologies. We establish its Hölder continuity in the generic setting with respect to the Wasserstein distance and its analyticity with respect to the total variation norm. In the non-invertible case, under appropriate assumptions, we obtain a characterization of uniform hyperbolicity via multi-cones and use it to establish a dichotomy between the analyticity and the discontinuity of the Lyapunov exponent. We also derive large deviations estimates and a central limit theorem for all of these models. While there are many interesting remaining open problems, our re sults attempt to provide an almost complete picture in the context of two dimensional random locally constant cocycles with finitely supported measu res.