Dissipative singular Sturm-Liouville operators are studied in the Hilbert space L-w(2) [a, b) (-infinity < a < b <= infinity), that the extensions of a minimal symmetric operator in Weyl's limit-point case. We construct a selfadjoint dilation of the dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and define its characteristic function in terms of the Titchmarsh-Weyl function of a selfadjoint operator. Finally, ill the case when the Titchmarsh-Weyl function of the selfadjoint operator is a meromorphic in complex plane, we prove theorems oil completeness of the system of eigenfunctions and associated functions of the dissipative Sturm-Liouville operators.