A development of Thom's catastrophe theory leads to a study the structural stability of some quasilinear PDEs. I apply this technique to study the Grad-Shafranov equation, which represents force-balance in a magnetically confined plasma with one ignorable coordinate. I found that critical equilibria could provide a necessary condition for confinement transitions in tokamaks, and physical arguments lead in the same direction as the mathematics. Similarly, critical equilibria may lead to loss of solution, which could be related to MHD instability and/or island formation.