We construct a space of boundary values of the minimal symmetric singular Dirac operator acting in the Hilbert space L-A(2) ([a, b); C-2) (- infinity < a < b <= infinity), and in Weyl's limit circle case. A description of all maximal dissipative, maximal accretive, selfadjoint, and other extensions of such a symmetric Dirac operator is given in terms of boundary conditions. We investigate two classes of maximal dissipative operators with separated boundary conditions, called 'dissipative at a' and 'dissipative at b'. In each of these cases we construct a selfadjoint dilation 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 maximal dissipative operator and define its characteristic function in terms of the Titchmarsh-Weyl function of a selfadjoint operator. Finally, we prove the theorems on completeness of the system of eigenvectors and associated vectors of the maximal dissipative Dirac operators.