The three-dimensional stability of the family of two-dimensional inviscid vortex patches discovered by Abrashkin & Yakubovich (Sov. Phys. Dokl. vol. 29, 1984, p. 370) is explored. Generally unsteady and non-uniform, these bounded regions of vorticity evolve freely in a surrounding irrotational flow. This family of solutions includes the Rankine circular vortex, Kirchhoff's ellipse, and freely rotating polygonal vortices as special cases. Taking advantage of their Lagrangian description, the stability analysis is carried out with the theory of local instabilities. It is shown that, apart from the Rankine vortex, these flows are three-dimensionally unstable. Background rotation or density stratification may however be stabilizing.