In this work, we explored the three-dimensional Kadomtsev-Petviashvili equation by the ${\varphi }^6$-model expansion method to generate precise solutions, such as periodic waves and singular structures, kink-type solitons, and dark solitons. The periodic solutions are the repeating patterns of a wave, and the singular solutions are the profiles that have unlimited amplitude at a certain point. The kink-type solitons depict sudden transitions, and the dark solitons represent local amplitude depressions in optical systems. Two- and three-dimensional plots were used to show the dynamic behavior and interactions of these solutions. The ${\varphi }^6$-model expansion technique is highly effective to study nonlinear systems, and can potentially be applied to fluid mechanics, plasma physics, and optical communication. The obtained results contribute to the understanding of the soliton propagation in the multi-dimensional systems.