We investigate the existence of two-stage and three-stage R-stable, P-stable, RL-stable, and dispersively enhanced diagonally implicit Runge-Kutta Nyström methods of orders three and four. We first show that a one-parameter family of two-stage third-order R-stable diagonally implicit methods exists, and that their dispersive order is at most four. From this we show that two-stage fourth-order R-stable, and third-order P-stable and RL-stable diagonally implicit methods do not exist. Next we show a two-parameter family of three-stage fourth-order R-stable diagonally implicit methods exists with dispersive order at most four, and that this family contains a one-parameter family of P-stable methods and a unique RL-stable. We also show that a one-parameter family of fourth-order diagonally implicit methods with dispersive order at least six exists, and that they are not R-stable. We present third- and fourth-order R-stable and P-stable methods with small principal truncation coefficients and discuss how these methods might be implemented in an efficient integrator.