Graphene as well as more generally Dirac solids constitute two-dimensional materials where the electronic flow is ultra-relativistic. When a Dirac solid is deposited on a different substrate surface with roughness, a local random potential develops through an inhomogeneous charge impurity distribution. This external potential affects profoundly the charge flow and induces a chaotic pattern of current branches that develops through focusing and defocusing effects produced by the randomness of the surface. An additional bias voltage may be used to tune the branching pattern of the charge carrier currents. We employ analytical and numerical techniques in order to investigate the onset and the statistical properties of carrier branches in Dirac solids. We find a specific scaling-type relationship that connects the physical scale for the occurrence of branches with the characteristic medium properties, such as disorder and bias field. We use numerics to test and verify the theoretical prediction as well as a perturbative approach that gives a clear indication of the regime of validity of the approach. This work is relevant to device applications and may be tested experimentally.