In a potential model for glueballs, with a dynamical gluon mass $m\ensuremath{\simeq}500$ MeV, the masses of the low-lying three-gluon glueballs (with ${J}^{\mathrm{PC}}={0}^{\ensuremath{-}+}, {1}^{\ensuremath{-}\ensuremath{-}}, \mathrm{and} {3}^{\ensuremath{-}\ensuremath{-}}$) are found to be approximately degenerate $\ensuremath{\simeq}4.8m\ensuremath{\simeq}2.4$ GeV. The ${\ensuremath{\eta}}_{c}\ensuremath{\rightarrow}\ensuremath{\eta}\ensuremath{\pi}\ensuremath{\pi}$, $\ensuremath{\psi}\ensuremath{\rightarrow}\ensuremath{\rho}\ensuremath{\pi}$ decays are used to extract mixing of three-gluon ${0}^{\ensuremath{-}+}$ and ${1}^{\ensuremath{-}\ensuremath{-}}$ glueballs with ${\ensuremath{\eta}}_{c}$ and $\ensuremath{\psi}$, respectively. The decays $\ensuremath{\psi}({\ensuremath{\psi}}^{\ensuremath{'}})\ensuremath{\rightarrow}h+X$ ($h=\ensuremath{\pi}\ensuremath{\pi},\ensuremath{\eta},{\ensuremath{\eta}}^{\ensuremath{'}}$) are suggested for experimentally studying the three-gluon continuum and three-gluon bound states. The three-gluon glueballs production rate via that mechanism as well as in $\mathrm{pp}$, $\ensuremath{\gamma}p$, and $\ensuremath{\gamma}\ensuremath{\gamma}$ collisions is estimated. The ${1}^{\ensuremath{-}\ensuremath{-}}$ decays dominantly into $\ensuremath{\rho}\ensuremath{\pi}$, but also has reasonable decay widths for ${K}^{*}\overline{K}$ channels. The ${0}^{\ensuremath{-}+}$ glueball state decays dominantly into $\mathrm{VV}$ ($V=\ensuremath{\rho},\ensuremath{\omega},\ensuremath{\varphi}$). Its other important decay modes are found to be $\ensuremath{\delta}\ensuremath{\pi}$, $V\ensuremath{\gamma}$, and $\ensuremath{\gamma}\ensuremath{\gamma}$.