  We prove that any normalized block sequence in a Schreier space $X_\xi$, of
arbitrary order $\xi<\omega_1$, admits a subsequence equivalent to a
subsequence of the canonical basis of some Schreier space. The analogous result
is proved for dual spaces to Schreier spaces. Basing on these results, we
examine the structure of strictly singular operators on Schreier spaces and
show that there are $2^\mathfrak{c}$ many closed operator ideals on a Schreier
space of any order, its dual and bidual space.
