This paper investigates the coupled nonlinear Hirota–Maccari system with the help of using an analytical approach, which is the extended sinh-Gordon equation expansion method (ShGEEM). Complex, solitary, and singular periodic traveling solutions are successfully gained to the nonlinear model considered. The constraint conditions that validate the existence of the reported soliton solutions are also given in a detailed manner. The two-dimensional (2D), three-dimensional, and contour graphs to some of the obtained solutions are presented via several computational programs. These simulations present deeper investigations about the wave distributions of the coupled nonlinear Hirota–Maccari system.