  We construct different types of quasiperiodically forced circle
homeomorphisms with transitive but non-minimal dynamics. Concerning the recent
Poincar\'e-like classification for this class of maps of Jaeger-Stark, we
demonstrate that transitive but non-minimal behaviour can occur in each of the
different cases. This closes one of the last gaps in the topological
classification. Actually, we are able to get some transitive quasiperiodically
forced circle homeomorphisms with rather complicated minimal sets. For example,
we show that, in some of the examples we construct, the unique minimal set is a
Cantor set and its intersection with each vertical fibre is uncountable and
nowhere dense (but may contain isolated points). We also prove that minimal
sets of the later kind cannot occur when the dynamics are given by the
projective action of a quasiperiodic SL(2,R)-cocycle. More precisely, we show
that, for a quasiperiodic SL(2,R)-cocycle, any minimal strict subset of the
torus either is a union of finitely many continuous curves, or contains at most
two points on generic fibres.
