  We study canonical bases for spaces of weakly holomorphic modular forms of
level 4 and weights in $\mathbb{Z}+\frac{1}{2}$ and show that almost all
modular forms in these bases have the property that many of their zeros in a
fundamental domain for $\Gamma_0(4)$ lie on a lower boundary arc of the
fundamental domain. Additionally, we show that at many places on this arc, the
generating function for Hurwitz class numbers is equal to a particular mock
modular Poincar\'{e} series, and show that for positive weights, a particular
set of Fourier coefficients of cusp forms in this canonical basis cannot
simultaneously vanish.
