Many-body superradiance in ordered atomic arrays is a phenomenon where atomic synchronization gives rise to a burst in photon emission. This superradiant burst only occurs if there is one -or just a few -dominant decay channels. By mapping the question of whether many-body superradiance occurs into a simple algebraic equation, we demonstrate that it exists in arrays of any dimensionality, as interference in photon emission leads to the preeminence of certain channels over others. In atomic chains superradiance occurs only below a critical interatomic distance, which we derive analytically. Increasing the array dimensionality leads to a critical distance that scales with system size: sublogarithmically for 2D, and seemingly faster than that for 3D lattices. We perform calculations for both infinite and finite systems, the latter by employing a highly-efficient algorithm with a computational complexity that scales only linearly with system size, which enables us to study very large arrays. Our results provide a guide to explore this many-body phenomenon in state-of-the art experimental setups.