This question arises in a somewhat naive form because I am largely unfamiliar with String Theory. I do know that it incorporates higher space dimensions where I shall take the overall dimensionality to be 10 in this question, for concreteness. Now the traditional Hawking-Penrose Singularity results apply to the the General Relativity manifold of 3+1 dimensions; with the 4D Schwarzchild solution providing an example of a Singularity and Black Hole.
So the question is: do singularities (and maybe associated Event Horizons) necessarily form in all 10 dimensions?
Examining this question for myself I see that this paper for mathematicians introduces an $N$ dimensional Schwarzchild metric and in theorem 3.15 an $N$ dimensional Hawking-Penrose singularity theorem. However this cannot answer directly to the intentions of the String theory models. For example it is mathematically possible to extend 4D Schwarzchild to 10D differently by adding a 6D Euclidean metric. So one question is whether this modified 10D Schwarzchild even meets the conditions for the $N$ dimensional Hawking-Penrose theorem. Although such a modification is not likely acceptable as a String Theory extension, it shows that we can consider some cases:
a) All 4D singularities / Event Horizons are actually 10 D ones.
b) Some/all 4D singularities / Event Horizons are "surface phenomena" in String theory - the underlying Bulk Volume is singularity free.
EDIT: Expressed a bit more formally this is saying that the String Theory has a singularity free solution $\Phi$ in 10D, but when $\Phi$ is restricted or reduced to 3+1D it is one of the known singular solutions of GR.
c) Some Singularities in String Theory Bulk (the 6D part) can arise without a corresponding 4D singularity (akin to a "deep earth earthquake" in 10D space-time, perhaps)?