  The consecutive pattern poset is the infinite partially ordered set of all
permutations where $\sigma\le\tau$ if $\tau$ has a subsequence of adjacent
entries in the same relative order as the entries of $\sigma$. We study the
structure of the intervals in this poset from topological, poset-theoretic, and
enumerative perspectives. In particular, we prove that all intervals are
rank-unimodal and strongly Sperner, and we characterize disconnected and
shellable intervals. We also show that most intervals are not shellable and
have M\"obius function equal to zero.
