Various kinds of splitting, or decomposition numerical methods have been playing important roles in the numerical solution of nonsingular partial differential equations due to their remarkable efficiency, simplicity and flexibility in computations as compared with their peers. Although the numerical strategy is still in its infancy for solving singular differential equation problems arising from many applications, explorations of the next generation decomposition schemes associated with different kinds of grid adaptations can be found in many recent publications. In this talk, we will discuss a few latest developments in the field. Key comments will be devoted to the direct solutions of degenerate singular reaction-diffusion equations and nonlinear sine-Gordon wave equations. Simulated numerical experimental results will be presented.