We have investigated numerically the stability and temporal evolution of two-dimensional self-gravitating galaxy rings with flat rotation curves. The initial equilibrium models are composed of a polytropic fluid characterized by various polytropic indices ( n = 0.5, 1, 1.5, 3, and 5) and embedded in a rigid scale-free logarithmic potential, but the fluid is evolved as an ideal gas with specific heat ratio γ = 5/3. Massless and very low-mass rings are unconditionally stable against both self-gravity driven modes and convective modes. As the self-gravity is increased, one of three nonaxisymmetric modes of instability soon appears, depending on the polytropic index and the radial extent: Jeans modes (J-modes) for n = 5 rings; Intermediate modes (I-modes) for 1 ≤ n ≤ 3 and for slender n = 0.5 rings; and gravity modes (g-modes), along with nonlinear I-modes, for extended n = 0.5 rings. The radial thickness of the orbiting fluid also influences the growth of these modes: more (less) extended rings require more (less) mass in order to develop an instability. If νp is the characteristic speed dictated by the potential and νo is the true orbital speed in the rings, then these instabilities operate for values of ν p/ν o ≤ 0.95-0.99 (ring-to-galaxy mass ratios approximately ≳0.11-0.02). At slightly higher mass ratios, strong unstable modes commonly cause a temporary breakup of the rings and the formation of new, lumpy but long-lived, ring structures. The entire process is completed within typically 3 orbits (J-modes) to 6-10 orbits (I-modes). Thus, neither nuclear rings in barred galaxies or in active galactic nuclei, nor (co)counter-rotating "cores" or polar rings in elliptical/S0 galaxies can safely accumulate matter through accretion and survive in a smooth form. In particular, some lumpy or distorted circumnuclear rings seen in barred galaxies may be due to the acting I-modes or J-modes that set in as soon as the accreted masses outgrow the relatively small critical value given above. Furthermore, the surprising absence of unstable g-modes from many models with n ≤ 1.5 which do not satisfy the Richardson stability criterion suggests that galaxy rings exhibit dynamical properties that cannot be identified by studying infinite cylindrical/annular models. Counter to intuition, the g-modes are absent because the shear and rotation rates are high in mildly compressible planar rings with flat rotation curves.