  In this paper, we use free field realisations of the A-type principal, or
Casimir, $W_N$ algebras to derive explicit formulae for singular vectors in
Fock modules. These singular vectors are constructed by applying screening
operators to Fock module highest weight vectors. The action of the screening
operators is then explicitly evaluated in terms of Jack symmetric functions and
their skew analogues. The resulting formulae depend on sequences of pairs of
integers that completely determine the Fock module as well as the Jack
symmetric functions.
