I personally have a question I want answered, although it is not limited to space curves singularities. All unibranch curve singularities have a natural degeneration to the monomial singularity with the same semigroup; in particular, plane curve singularities with more than one Puiseux pair will degenerate to space(or higher) curve singularities. For example, the plane curve singularity $k[t^4,t^6 + t^7]$ degenerates to the space curve singularity $k[t^4, t^6, t^{13}]$.

I would like to understand how the (local contribution to the) relative compactified Jacobian degenerates in this family. The question I really want answered is: which torsion free sheaves on the central fibre deform out? In particular, it is known that the compactified Jacobian is irreducible if and only if the singularity is planar. Does the deformation to the monomial curve select out a component of the compactified Jacobian of the monomial curve singularity and, if so, is this component homeomorphic to the compactified Jacobian of the original planar curve singularity?

Let me also mention an obvious question following upon the work of Campillo, Delgado, and Gusein-Zade, who observe that the semigroup of a plane curve singularity "is" its Alexander polynomial. So: what is the topological meaning of the semigroup of a space curve singularity? An answer would of course bear upon the question of which such semigroups occur.