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It

Edited. As Richard points out this is arguable not really an answer, but rather a comment, that most questions on even for plane singularities have been answered. For example 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

http://arxiv.org/abs/1003.1568

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It is arguable that most questions on plane singularities have been answered. For example there is a completely (new and) open conjecture in the following article

The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link

http://arxiv.org/abs/1003.1568