  Let ${\mathfrak{g}}$ be a complex semisimple Lie algebra with Borel
subalgebra ${\mathfrak{b}}$ and corresponding nilradical ${\mathfrak{n}}$. We
show that singular Whittaker modules $M$ are simple if and only if the space
$\hbox{Wh}\,M$ of Whittaker vectors is $1$-dimensional. For arbitrary locally
${\mathfrak{n}}$-finite ${\mathfrak{g}}$-modules $V$, an immediate corollary is
that the dimension of $\hbox{Wh}\,V$ is bounded by the composition length of
$V$.
