  We study Riemannian foliations whose transverse Levi-Civita connection
$\nabla$ has special holonomy. In particular, we focus on the case where
$Hol(\nabla)$ is contained either in SU(n) or in Sp(n). We prove a Weitzenbock
formula involving complex basic forms on K\"ahler foliations and we apply this
formula for pointing out some properties of transverse Calabi-Yau structures.
This allows us to prove that links provide examples of compact simply-connected
contact Calabi-Yau manifolds. Moreover, we show that a simply-connected compact
manifold with a K\"ahler foliation admits a transverse hyper- K\"ahler
structure if and only if it admits a compatible transverse hyper-Hermitian
structure. This latter result is the "foliated version" of a theorem proved by
Verbitsky. In the last part of the paper we adapt our results to the Sasakian
case, showing in addition that a compact Sasakian manifold has trivial
transverse holonomy if and only if it is a compact quotient of the Heisenberg
Lie group.
