<p>A stable meshless method by constructing novel fractional spaces and fractional Chebyshev polynomials dense sets is developed. The proposed approach could excellently resolve 2D nonlinear Riesz space distributed order reaction-diffusion equations with non-smooth solutions. The stable discretization schemes and predictor-corrector method are applied to deal with the equations. We propose novel and important fractional spaces from a new perspective, which solve the problem of selecting solution spaces. In addition, we construct the fractional dense subsets or meshless shape functions, which can approximate Riesz space distributed order derivatives with higher accuracy and improve traditional moving least squares meshless method. The approach suitably divides the exact solution of the equation into four parts and successfully deals with the weak singularities of the solution at multiple endpoints and edges of the space. This approach novelly improves the smoothness of the exact solution, which is different from the previous traditional smooth transformation. Importantly, the convergence order and stability of the meshless approach are analyzed. Eventually, some numerical examples demonstrate the effectiveness and practicability of our method, which could approximate the non-smooth exact solutions with high precision.</p>