<p>In this paper, we investigate the (3+1)-dimensional extended Estevez–Prada equation, which describes the propagation of nonlinear waves in high-dimensional media. The primary objective is to derive a broad class of exact analytical solutions that capture complex wave phenomena such as stable soliton formation and wave interactions in real-world complex media, including plasmas, optical fibers, and fluid dynamics. To achieve this, we apply the Modified Extended Direct Algebraic Method, which generates several families of solutions expressed in terms of hyperbolic, rational, and trigonometric functions. These solutions include kink-type solitons, dark solitons, periodic oscillating waves, kink-antikink interactions, and many other distinct wave shapes. A key finding is that our approach yields moving singularities, periodic trains of infinite spikes, and dark soliton U-shaped depressions, which have not been reported in previous studies of this equation. The partial differential equation is first transformed into an ordinary differential equation via an appropriate wave transformation, after which an algebraic system is constructed and solved to obtain explicit solutions for various families. The successful application of this method confirms its effectiveness for high-dimensional nonlinear evolution equations. We also verify the Painlevé properties, demonstrating the absence of movable critical singularities and thereby supporting the existence of the obtained explicit solutions. Graphical representations, including two-dimensional, contour, and three-dimensional plots, illustrate how the governing parameters affect the amplitude, width, and propagation of the waves. Overall, these results establish the Modified Extended Direct Algebraic Method as an effective and reliable tool for high-dimensional nonlinear evolution equations, providing deep insights into complex wave phenomena in mathematical physics.</p>