The Material Point Method (MPM) is widely utilized for simulating boundary value problems in continuum mechanics, offering unique advantages such as eliminating mesh distortions in large-deformation problems. This study investigates and compares the application of linear shape functions, enhanced by the grid-shift technique, with quadratic B-splines in MPM with respect to the stress oscillations of the solutions. Linear shape functions are computationally efficient but suffer from the cell-crossing error due to discontinuities in gradient fields across element boundaries. The grid-shift technique mitigates this by introducing random shifts to the computational background grid, achieving smoother stress fields without additional computational costs. Quadratic B-splines, on the other hand, naturally provide \(C^1\) -continuity, reducing cell-crossing errors and producing smoother results at the cost of an increased computational overhead.