We analyze the formation of stripe-like and droplet-like superstructures in a plane with classical magnetic moments forming a square lattice, which is relevant, in particular, to a two-dimensional array of π-rings. Each magnetic and orbital moment associated with a single π-ring is oriented perpendicular to the plane. We describe such arrays in the framework a two-dimensional Ising model with both nearest-neighbor and next-nearest-neighbor (diagonal) interactions. Such competing interactions introducing a strong frustration provide a plausible description for a dipole-dipole type of interaction. The model was studied with the use of several methods including exact diagonalization for small clusters, transfer matrix technique, as well as the Monte Carlo simulations for large lattices. The complete phase diagram resulting from this study includes, together with the conventional antiferromagnetic Ising and stripe phases, an additional glassy phase characterized by the formation of large-scale superstructures.