This talk presents examples of a remarkable stabilizing phenomenon in fluid stability problems. The 3D incompressible Euler equation can blow up in a finite time. Even small data would not help. But when the 3D Euler is coupled with the non-Newtonian stress tensor, as in the Oldroyd-B model, small smooth data always lead to global and stable solutions. The 3D Navier-Stokes equation with dissipation in only one direction is not known to always have small global solutions. However, when it is coupled with the magnetic field in the magnetohydrodynamic system, solutions near a background magnetic field are always global in time. The magnetic field stabilizes the fluid. Mathematically this stabilizing phenomenon boils down to wave structures hidden in the systems governing perturbations around physically relevant steady states.