  Finite discrete subgroups of $U(3)$ as possible flavour symmetries $G_f$ for
a massless neutrinos with predictive mixing angles are studied. This is done by
assuming that a residual symmetry $S_\nu$ appropriate for describing a massless
neutrino is contained in $G_f$. It is shown that all the groups $G_f$ admitting
three dimensional faithful irreducible representation and generated from a
specific set of $3\times 3$ matrices imply only one of the three flavour
compositions for the massless state namely, unmixed, maximally mixed with equal
probabilities and bimaximally mixed with probabilities (0,1/2,1/2) and their
permutations. This result holds irrespective of the order of $G_f$ and the
choice of $S_{\nu}$ within it. All of these lead to unfavourable leading order
prediction for the solar mixing angle. Neutrino mixing pattern is then
numerically investigated in case of subgroups of $U(3)$ with order less than
512 and it is found that only one of these can lead to a massless neutrino and
leading order predictions for all the mixing angles close to their experimental
values. Ways to correct for the solar angle prediction are proposed and two
concrete examples giving the observed mixing pattern are discussed.
