As far as I am aware, no stable orbits can exist inside of the event horizon. There is a region just outside of the event horizon (called the photon sphere) where the closest stable orbit can exist. This is about 50% larger than the diameter of the event horizon itself. It is at this point where objects travelling at the speed of light (i.e. photons) can enter a stable orbit. Any closer than that, and you'd have to exceed the speed of light to keep your orbit. Since that isn't possible, you fall in.

It seems to me that this might lead to some really interesting effects. If a photon were to scatter off a particle of matter at the right position, then some of its wave function would be scattered in such a manner as to lock it into such an orbit. Should we expect every long-lived black hole to have orbiting photons in stable orbits?

If photons could be scattered into stable orbits, then should it also be possible for matter entering the black hole to scatter them out again?? Are black holes really white?

And finally, this interacts with another question in this forum -- if a photon is in a stable orbit around a black hole, and the length of the orbit is an integral number of wavelengths (analogue of Bohr hydrogen atom model) then its wavefunction is a stationary state. Is this a case of a stationary photon?