For plane curve singularities most questions have been answered, in large part due to the NewtonPuiseux expansion. I've heard that there are a number of open problems regarding space curve singularities but have not found any clear statements. (One problem or set of problems seems to be around the characterization of which semigroups can arise.) Could someone point me to some of these problems? I am primarily interested in local singularities, over the complex numbers or reals, and to start with, unibranched singularities.

Edited. As Richard points out this is not really an answer, but rather a comment, that even for plane sinuglarities there are still some open questions. There is a facinating, completely (new and) open conjecture in the following article The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link 


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 GuseinZade, 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. 


I think the following is still open: "Does there exist an isolated space curve singularity that is rigid?" 

