  We construct a family $\{\Phi_t\}_{t\in[0,1]}$ of homeomorphisms of the
two-torus isotopic to the identity, for which all of the rotation sets
$\rho(\Phi_t)$ can be described explicitly. We analyze the bifurcations and
typical behavior of rotation sets in the family, providing insight into the
general questions of toral rotation set bifurcations and prevalence. We show
that there is a full measure subset of $[0,1]$, consisting of infinitely many
mutually disjoint non-trivial closed intervals, on each of which the rotation
set mode locks to a constant polygon with rational vertices; that the generic
rotation set in the Hausdorff topology has infinitely many extreme points,
accumulating on a single totally irrational extreme point at which there is a
unique supporting line; and that, although $\rho(t)$ varies continuously with
$t$, the set of extreme points of $\rho(t)$ does not. The family also provides
examples of rotation sets for which an extreme point is not represented by any
minimal invariant set, or by any directional ergodic measure.
