Numerical investigations support the existence of axisymmetric stationary Einstein--Vlasov bodies; that is, solutions with finite mass and compact support. A conjecture going back to Zel'dovitch, Novikov, and others states that along a parameterized sequence of such bodies, the members become unstable at the maximum of the binding energy. In this talk I will present results of our recent numerical investigations into this conjecture in the axisymmetric setting, as well as results on the dynamics of perturbed stationary states.