  We use crystals of tableaux and descent compositions to understand the
decomposition of Schur functions $s_\lambda$ into Gessel's fundamental
quasisymmetric functions $F_\alpha$. The connected crystal of tableaux
$B(\lambda)$, associated to $s_\lambda$, is shown to be partitionned into a
disjoint union of connected induced subgraphs $B(T_\alpha)$ corresponding to
the $F_\alpha$'s.
  We show that these subgraphs, which we call quasicrystals, are isomorphic (as
graphs) to specific crystals of tableaux. This allows us to give a formula for
the number of tableaux of shape $\lambda$ and maximal entry $n$. We also use
this setting to give a constructive proof of a combinatorial formula for Kostka
numbers $K^\lambda_\mu$. We study the position of the quasicrystals within the
crystal $B(\lambda)$, and show that they appear in dually positionned pairs,
with the crystal anti-automorphism between them being given by a generalization
of Sch\"utzenberger's evacuation. We introduce the notion of skeleton of the
crystal $B(\lambda)$ given by replacing each subgraph $B(T_\alpha)$ by the
associated standard tableau of shape $\lambda$. We conjecture that its graph
includes the dual equivalence graph for $\lambda$, introduced by Assaf, and
that its subgraphs of tableaux with fixed number of descents have particular
structures. Finally, we describe applications to plethysm, among which we give
an algorithm to express any symmetric sum of fundamental quasisymmetric
functions into the Schur basis, whose construction gives insight into the
relationship between the two basis.
