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The aim of this paper is the accurate numerical study of the Kadomtsev–Petviashvili (KP) equation. In particular, we are concerned with the small dispersion limit of this model, where no comprehensive analytical description exists so far. To this end, we first study a similar highly oscillatory regime for asymptotically small solutions, which can be described via the Davey–Stewartson system. In a second step, we investigate numerically the small dispersion limit of the KP model in the case of large amplitudes. Similarities and differences to the much better studied Korteweg–de Vries situation are discussed as well as the dependence of the limit on the additional transverse coordinate. [ABSTRACT FROM AUTHOR]