  We introduce a notion of rigid local system on the comple- ment of a plane
curve $Y$, which relies on a canonical Waldhausen de- composition of the Milnor
sphere associated to $Y$. We show that when $Y$ is weigthed homogeneous this
notion is deeply related to the classical notion of rigidity on the Riemann
sphere. We construct large families of rigid local systems on the complement of
weighted homogeneous plane curves and show that the corresponding $D$-modules
are generated by `special' multivalued holomorphic functions.
