  Poincare's classification of the dynamics of homeomorphisms of the circle is
one of the earliest, but still one of the most elegant, classification results
in dynamical systems. Here we generalize this to quasiperiodically forced
circle homeomorphisms, which have been the subject of considerable interest in
recent years. Herman already showed two decades ago that a unique rotation
number exists for all orbits in the quasiperiodically forced case. However,
unlike the unforced case, no a priori bounds exist for the deviations from the
average rotation. This plays an important role in the attempted classification,
and in fact we define a system as regular if such deviations are bounded and
irregular otherwise. For the regular case we prove a close analogue of
Poincare's result: if the rotation number is rationally related to the rotation
rate on the base then there exists an invariant strip (an appropriate analogue
for fixed or periodic points in this context), otherwise the system is
semi-conjugate to an irrational translation of the torus. In the irregular
case, where neither of these two alternatives can occur, we show that the
dynamics are always topologically transitive.
