The standard Material Point Method (MPM) [1], originally developed as an extension of FLIP from fluid to solid mechanics, may exhibit overly restrictive stable time-step limits in explicit dynamic simulations. Instability is commonly observed in cells that are only partially filled. Recent single-step spectral analyses [2] have shown that the stability time-step limit is highly sensitive to the cell-wise particle center of mass and may reduce to zero as particles approach cell boundaries. Moreover, a distinct ``ringing'' instability may develop over multiple time steps when grid-represented fields are under-resolved relative to particle-represented fields; this instability can be mitigated by introducing smoother grid-represented fields [3]. Building on the stability framework developed in [2], this presentation first reviews the closed-form stability conditions obtained from single-step spectral analysis and the identification of dominant stability-controlling factors using a one-dimensional oscillating rod model. The associated stabilization coefficient for critical time-step relaxation and velocity-gradient reconstruction scheme for ringing mitigation are then summarized. Finally, preliminary extensions of the framework to two-dimensional simulations on regular Cartesian grids are presented to demonstrate its applicability and limitations in multi-dimensional problems.