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We report on a search for the radiative decay of $\ensuremath{\Upsilon}(1\mathrm{S})$ to the pseudoscalar mesons $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$ in $(21.2\ifmmode\pm\else\textpm\fi{}0.2)\ifmmode\times\else\texttimes\fi{}{10}^{6}$ $\ensuremath{\Upsilon}(1\mathrm{S})$ decays collected with the CLEO III detector at the Cornell Electron Storage Ring. The $\ensuremath{\eta}$ meson was reconstructed in the three modes $\ensuremath{\eta}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma}$, $\ensuremath{\eta}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$, or $\ensuremath{\eta}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$. The ${\ensuremath{\eta}}^{\ensuremath{'}}$ meson …