We can define the stability of a vortex by perturbing it in some way and seeing if that perturbation grows or decays.
For an initially circular vortex ring, we could add a sinusoidal perturbation to its shape and see if the amplitude of the sinusoid grows. It has been shown that vortex rings are unstable to such a sinusoidal perturbation.
For the bathtub vortex you ask about, it is itself the result of an instability of the flow draining down a plug. Water draining from a bath tub could drain without any swirl or vortex, the flow velocity could be entirely radial towards the plughole. Indeed, for low enough $(\lessapprox50)$ Reynolds numbers that is what happens.
However, for higher Reynolds numbers the base flow is unstable to a swirl perturbation and the flow starts to rotate faster and faster, forming the familiar bathtub vortex. More detail can be found in this paper.