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We consider contact 3-manifolds that arise in a canonical way as links of surface singularities and try to get information about the singularity from the topological invariants of the contact structure. We will focus on open book decompositions, which encode contact structures via fibered links (due to work of Giroux). A contact structure is called planar if it admits an open book with fibers of genus 0. We show that a hypersurface singularity gives a planar contact structure if and only if the singularity is the simplest possible (namely of type A_n). For general normal surface singularity, the canonical contact structure on a link is planar if and only if the singularity is minimal (minimality means a restrictive condition on the resolution graph). Our proofs use topology of Lefschetz fibrations; I will give the necessary background in the graduate seminar and explain the proofs in this talk. Time permitting, I will discuss further corollaries and some connections of our results to deformations and smoothings of singularities. (Joint work with P.Ghiggini and M. Golla.)