  Pulsar timing arrays have found evidence for a low-frequency gravitational
wave background (GWB). Assuming the GWB is produced by supermassive black hole
binaries (SMBHBs), the next gravitational wave (GW) signals astronomers
anticipate are Continuous Waves (CWs) from single SMBHBs and their associated
GWB anisotropy. The prospects for detecting CWs and anisotropy are highly
dependent on the astrophysics of SMBHB populations. Thus, information from
single sources can break degeneracies in astrophysical models and place much
more stringent constraints than the GWB alone. We simulate and evolve SMBHB
populations, model their GWs, and calculate their anisotropy and detectability.
We investigate how varying components of our semi-analytic model, including the
galaxy stellar mass function, the SMBH--host galaxy relation
($M_\mathrm{BH}$--$M_\mathrm{bulge}$), and the binary evolution prescription
impact the expected detections. The CW occurrence rate is greatest for few
total binaries, high SMBHB masses, large scatter in
$M_\mathrm{BH}$--$M_\mathrm{bulge}$, and long hardening times. The occurrence
rate depends most on the binary evolution parameters, implying that CWs offer a
novel avenue to probe binary evolution. The most detectable CW sources are in
the lowest frequency bin for a 16.03-year PTA, have masses from
$\sim\!\!10^9-10^{10}\mathrm{M}_\odot$, and are $\sim\!\!1$ Gpc away. The level
of anisotropy increases with frequency, with the angular power spectrum over
multipole modes $\ell$ varying in low-frequency $C_{\ell>0}/C_0$ from
$\sim\!\!5\times 10^{-3}$ to $\sim\!\!2\times10^{-1}$, depending on the model;
typical values are near current upper limits. Observing this anisotropy would
support SMBHB models for the GWB over cosmological models, which tend to be
isotropic.
