We investigate the landscape of many-body memories: families of local non-equilibrium dynamics that retain information about their initial conditions for thermodynamically long time scales, even in the presence of arbitrary perturbations. In two dimensions, the only well-studied memory is Toom's rule. Using a combination of rigorous proofs and machine learning methods, we show that the landscape of 2D memories is in fact quite vast. We discover memories that correct errors in ways qualitatively distinct from Toom's rule, have ordered phases stabilized by fluctuations, and preserve information only in the presence of noise. Taken together, our results show that physical systems can perform robust information storage in many distinct ways, and demonstrate that the physics of many-body memories is richer than previously realized. Interactive visualizations of the dynamics studied in this work are available at https://memorynca.github.io/2D.