We explore the discovery prospect of the $pp$, $e^-p$ and $e^+ e^-$ colliders at the LHC, FCC-he and CLIC for model-independent

sensitivity estimates on the dipole moments of the $\tau$-lepton $\tilde{a}_\tau$ and $\tilde{d}_\tau$. We consider the channels

$pp \to p\tau ^+ \tau^- \gamma p$, $e^-p \to e^- \tau^+ \tau^- \gamma p$ and $e^+e^- \to e^+ \tau ^+ \tau^- \gamma e^-$. Bounds

on the dipole moments of the $\tau$-lepton are obtained: $\tilde a_\tau= (-0.00674, 0.00658)$,

$ |\tilde{d}_\tau({\rm ecm})|= 3.6925\times 10^{-17}$ [LHC]; $\tilde a_\tau= (-0.00265, 0.00246)$,

$ |\tilde{d}_\tau({\rm ecm})|= 1.4379\times 10^{-17}$ [FCC-he] and $\tilde a_\tau= (-0.00128, 0.00105)$,

$ |\tilde{d}_\tau({\rm ecm})|= 6.4394\times 10^{-18}$ [CLIC]. The sensitivity of CLIC show a potential advantage compared to those

from LHC and FCC-he.