  Let $g$ be a Hecke cusp form of half-integral weight, level $4$ and belonging
to Kohnen's plus subspace. Let $c(n)$ denote the $n$th Fourier coefficient of
$g$, normalized so that $c(n)$ is real for all $n \geq 1$. A theorem of
Waldspurger determines the magnitude of $c(n)$ at fundamental discriminants $n$
by establishing that the square of $c(n)$ is proportional to the central value
of a certain $L$-function. The signs of the sequence $c(n)$ however remain
mysterious. Conditionally on the Generalized Riemann Hypothesis, we show that
$c(n) < 0$ and respectively $c(n) > 0$ holds for a positive proportion of
fundamental discriminants $n$. Moreover we show that the sequence $\{c(n)\}$
where $n$ ranges over fundamental discriminants changes sign a positive
proportion of the time. Unconditionally, it is not known that a positive
proportion of these coefficients are non-zero and we prove results about the
sign of $c(n)$ which are of the same quality as the best known non-vanishing
results. Finally we discuss extensions of our result to general half-integral
weight forms $g$ of level $4 N$ with $N$ odd, square-free.
