Experiments are conducted to generate progressive wave fields in deep water with two-dimensional surface patterns for which two parameters are systematically varied: (i) the aspect ratio of the cells comprising the surface patterns and (ii) a measure of nonlinearity of the input wave field. The goal of these experiments is to determine whether these patterns persist, what their main features are, whether standard models of waves describe these features, and whether there are parameter regimes in which the patterns are stable. We find that in some parameter regimes, surface patterns in deep water do persist with little change of form during the time of the experiment. In other parameter regimes, particularly for large-amplitude experiments, the patterns evolve more significantly. We characterize the patterns and their evolutions with a list of observed features. To describe the patterns and features, we consider two models: ($a$) the standard ($2+1$) nonlinear Schrödinger equation and ($b$) coupled nonlinear Schrödinger equations for two interacting wavetrains. Exact solutions of these models provide qualitative explanations for many of the observed features.