  Schreier graphs of the actions of Thompson's group $F$ on the orbits of all
points of the unit interval and of the Cantor set with respect to the standard
generating set $\{x_0,x_1\}$ are explicitly constructed. The closure of the
space of pointed Schreier graphs of the action of $F$ on the orbits of dyadic
rational numbers and corresponding Schreier dynamical system are described. In
particular, we answer the question of Grigorchuk on the Cantor-Bendixson rank
of the underlying space of the Schreier dynamical system in the context of $F$.
As applications we prove that the pointed Schreier graphs of points from
$(0,1)$ are amenable, have infinitely many ends, and are pairwise
non-isomorphic. Moreover, we prove that points $x,y\in(0,1)$ have isomorphic
non-pointed Schreier graphs if and only if they belong to the same orbit of
$F$.
