In the conformal diagram for the Reissner-Nordström solution (included below), the inner horizon of the black hole leads to a wormhole which has a singularity, but it is possible to exit it through a white hole in another universe. I've read several sources that call this wormhole "unstable", but without any justification for this fact. What makes it unstable?
I can give you only a partial account because this is an area I'm only partially familar with. If you're feeling brave there is a rigorous analysis of the physics involved in this paper.
The Reissner-Nordström geometry is perfectly stable as long as no extra matter is present, so in the conformal diagram you've included there is no instability. The trouble is that matter falling in from our universe meets matter falling in from the other universe at the inner horizon and produces an effect called mass inflation. This produces a contribution to the stress-energy tensor that makes the Reissner-Nordström geometry unstable.
For completeness I should note that this is all interesting but physically irrelevant, because the Reissner-Nordström geometry describes an eternal charged black hole and no such object can exist in the real universe. The black hole must have a finite age due to the Big Bang and cannot exist forever due to Hawking evaporation, so no wormholes and parallel universes actually exist. It was a sad day when I realised this was the case.