Yes they do. In fact, such standing wave can form intensity pattern in amplifier, therefore many interesting effects may happen, e.g., cross-gain modulation, reflections due to standing wave pattern, etc.

But ring lasers support only traveling wave patterns, then how are standing wave patterns supported?

Laser cavity is a confined structure so it can support standing wave.

In ring lasers clockwise and counterclockwise propagation both exist with the same speed. They interfere and together forms a standing wave.

And are these patterns independent of the laser medium used i.e. solid, liquid or gaseous?

The wave number of the beating waves depends on the refractive index of the medium. Therefore the standing wave pattern formed by the beating waves will depend on the medium.

Standing waves can form in a ring cavity in general. However, we usually find non-reciprocal optical components in a ring cavity. These components will generate losses that are not balanced for the clockwise and anticlockwise propagating signals and will consequently prevent the formation of standing waves. In fact, one usually insert an optical isolator in such a cavity to force clockwise operation and avoid the detrimental effects mentionned by M. Jiao (cross gain saturation, hole-burning, reflections, etc.)..

And during interference does the clockwise and counter-clockwise lose their traveling wave nature?

Some other effects have to be added to the equation. First of all these patterns are, in principle, not 'standing', as the phase relationship between clockwise and counterclockwise modes is not fixed, and coherence length is finite. Secondly, however, because of mode coupling, e.g., by scatterers and/or smal reflections, these two modes can actually be fixed in phase with respect to eachother.

Why are ring resonators then specifically called Traveling Wave resonators ?