Singular systems of linear forms were introduced by Khintchine in the 1920s, and it was shown by Dani in the 1980s that they are in one-to-one correspondence with certain divergent orbits of one-parameter diagonal groups on the space of lattices. We give a (conjecturally sharp) upper bound on the Hausdorff dimension of singular systems of linear forms (equivalently the set of points with divergent trajectories). This extends work by Cheung, as well as by Chevallier and Cheung, on the vector case. For a diagonal action on the space of lattices, we also consider the relation of the entropy of an invariant measure to its mass in a fixed compact set. Our technique is based on the method of integral inequalities developed by Eskin, Margulis, and Mozes. This is a joint work with D. Kleinbock, E. Lindenstrauss, and G. A. Margulis.