In this study, we investigate the nonlinear Zoomeron equation by using the extended rational sin − cos and sinh − cosh methods. We successfully constructed some important solutions such as singular periodic wave, periodic wave, topological, and singular soliton solutions. Using suitable parameter values, we present the numerical simulations to some of the obtained results via the 2- and 3-dimensional graphs. The methods used in this study are efficient and powerful mathematical tools that can be applied in reporting various wave solutions to different nonlinear physical models.