The production of final state photons in hadronic $Z$-boson decays
can be used to study the quark-to-photon fragmentation function
$D_{q\to \gamma}(z,\mu_{F})$. Currently, two different observables
are used at LEP to probe this function: the `photon' +~1 jet rate
and the inclusive photon energy distribution. We outline the results
of a calculation of the `photon' +~1 jet rate
at fixed ${\cal O}(\alpha \alpha_{s})$, which yield
a next-to-leading order determination of the quark-to-photon
fragmentation function $D_{q\to \gamma}(z,\mu_{F})$. The resulting
predictions for the isolated photon rate and the inclusive photon
spectrum at the same, fixed order, are found to be in good agreement
with experimental data. Furthermore, we outline the main features of
conventional approaches using parameterizations of the resummed
solutions of the evolution equation and
point out deficiencies of these currently available
parameterizations in the large $z$-region. We
finally demonstrate that the ALEPH data on the
`photon' +~1 jet rate are able to
discriminate between different parameterizations
of the quark-to-photon fragmentation function, which are equally
allowed by the OPAL photon energy distribution data.
