  We prove that if a contact 3-manifold admits an open book decomposition of
genus 0, a certain intersection pattern cannot appear in the homology of any of
its minimal symplectic fillings, and moreover, fillings cannot contain
symplectic surfaces of positive genus. Applying these obstructions to canonical
contact structures on links of normal surface singularities, we show that links
of isolated singularities of surfaces in the complex 3-space are planar only in
the case of $A_n$-singularities. In general, we characterize completely planar
links of normal surface singularities (in terms of their resolution graphs);
these singularities are precisely rational singularities with reduced
fundamental cycle (also known as minimal singularities). We also establish
non-planarity of tight contact structures on certain small Seifert fibered
L-spaces and of contact structures arising from the Boothby--Wang construction
applied to surfaces of positive genus. Additionally, we prove that every
finitely presented group is the fundamental group of a Lefschetz fibration with
planar fibers.
