See the figure between pages 14 and 15 from the link below.
There is a weak curvature singularity at the inner (Cauchy) horizon of a rotating black hole. Seminal work on this was done by Poisson and Israel, and this work was continued by Ori. See
http://physics.technion.ac.il/~school/Amos_Ori.pdf ,
particularly pages 15, starting at "Consequence to the curvature singularity at the IH: (IH = Inner Horizon), 16, and 24. On page 24, Ori says that classical general relativity cannot predict what happens inside the inner horizon,
For Novikov's take on this, see
http://arxiv.org/abs/gr-qc/0304052.
Roughly, if components of g (the metric) are continuous but "pointy" (like the absolute value function), then first derivatives of g have step diiscontinuities (like the Heaviside step function), and second derivatives of g (used in the curvature tensor) are like Dirac delta functions. If a curvature singularity blows up like a Dirac delta function, then integration produces only a finite contribution to the tidal deformation of an object, which, if the object is robust enough, it can withstand.