  We describe the emergence of superfluid phases of ultracold dipolar fermions
in optical lattices for two-dimensional systems. Considering the many-body
screening of dipolar interactions at intermediate and larger filling factors,
we show that several superfluid phases with distinct pairing symmetries
naturally arise in the singlet channel: local s-wave $(sl)$, extended s-wave
$(se)$, d-wave $(d)$ or time-reversal-symmetry breaking $(sl + se \pm
id)$-wave. We obtain the temperature versus filling factor phase diagram and
show that d-wave pairing is favored near half-filling, that $(sl + se)$-wave is
favored near zero or full filling, and that time-reversal-breaking $(sl + se
\pm id)$-wave is favored in between. The inclusion of a harmonic trap reveals
that a sequence of phases can coexist in the cloud depending on the filling
factor at the center of the trap. Most notably in the spatial region where the
$(sl + se \pm id)$-wave superfluid occurs, spontaneous currents are generated,
and may be detected using velocity sensitive Bragg spectroscopy.
