Since toroidal planet are hypothetically possible, albeit very unlikely to actually exist, can we extend this idea to black holes? Would these unstable bodies work atleast on paper?
First, let's clarify that there are unphysical, coordinate dependent singularities and there are physical, coordinate independent singularities. An event horizon is an unphysical singularity, while the ring singularity of the Kerr metric is a physical singularity.
Since toroidal planet are hypothetically possible, albeit very unlikely to actually exist, can we extend this idea to black holes? Would these unstable bodies work at least on paper?
Yes, this is a mathematically sound concept. Axisymmetric black hole solutions (i.e. Kerr) that are toroidal near the ring singularity was first pointed out by Kip Thorne in 1975 (available for free here). The event horizon can be toroidal, not the physical singularity itself. These toroidal horizons are not necessarily in violation of cosmic censorship. But one must be careful as it can apparently lead one to slip into quackery. More recently, new solutions have been found of rotating toroidal black holes.
Although there is no observational evidence to support this idea, it was recently reported in a numerical relativity simulation that the event horizon of a binary black hole merger can be temporarily toroidal, as was predicted earlier, and which does not violate topological censorship. This is very interesting, as it implies that a toroidal black hole horizon may be observable.