Mesh-based numerical methods commonly used in computational mechanics often face limitations related to mesh generation, refinement, and numerical robustness, particularly when higher-order differential equations are involved. As an alternative, meshless methods have attracted increasing attention due to their flexibility and ease of implementation. Among these approaches, Radial Basis Function (RBF) methods provide an efficient framework for approximating field variables without requiring spatial discretization. This paper presents a meshless RBF-based formulation for the numerical solution of fourth-order ordinary differential equations arising in structural stability problems. Two representative benchmark problems are considered: lateral–torsional buckling of steel beams and pure torsional buckling of steel columns. These problems are governed by well-established differential equations and are suitable for assessing the accuracy and reliability of the proposed approach. The governing equations are solved using a collocation strategy with globally supported radial basis functions, allowing direct enforcement of boundary conditions and avoiding mesh-related issues. The numerical results are compared with classical analytical solutions and with reference results obtained using the Finite Difference Method. The comparisons show very good agreement, with low relative errors even when a small number of collocation points is employed. The results indicate that the proposed RBF formulation is accurate, stable, and computationally efficient for stability problems governed by higher-order differential equations. The study demonstrates the potential of meshless RBF-based techniques as a robust alternative to traditional discretization methods for higher-order problems in computational solid and structural mechanics.