  Earlier Monte-Carlo calculations on the dissipative two-dimensional XY model
are extended in several directions. We study the phase diagram and the
correlation functions when dissipation is very small, where it has properties
of the classical 3D-XY transition, i.e. one with a dynamical critical exponent
$z=1$. The transition changes from $z=1$ to the class of criticality with $z
\to \infty$ driven by topological defects, discovered earlier, beyond a
critical dissipation. We also find that the critical correlations have
power-law singularities as a function of tuning the ratio of the kinetic energy
to the potential energy for fixed large dissipation, as opposed to essential
singularities on tuning dissipation keeping the former fixed. A phase with
temporal disorder but spatial order of the Kosterlitz-Thouless form is also
further investigated. We also present results for the transition when the
allowed Caldeira-Leggett form of dissipation and the allowed form of
dissipation coupling to the compact rotor variables are both included. The
nature of the transition is then determined by the Caldeira-Leggett form.
