Branching fractions of $\eta$ and $\eta’(958)$ anomalous decays at LHCb 2025 Branching fractions of $\eta$ and $\eta’(958)$ anomalous decays are measured using $pp$ collision data at a center-of-mass energy of $\sqrt{s}=13\,\mathrm{TeV}$ collected by the LHCb experiment and corresponding to an integrated luminosity of $5.4\,\mathrm{fb^{-1}}$. The measured values and upper limits at the 90% confidence level are \begin{equation} \mathcal{B}(\eta\to\mu^+\mu^-) = (5.30\pm0.14_{\text{(stat)}}\pm0.18_{\text{(syst)}}\pm0.26_{\phi})\times10^{-6} \nonumber, \end{equation} \begin{equation} \mathcal{B}(\eta’(958)\to\mu^+\mu^-)<1.94\times10^{-7} \nonumber, \end{equation} \begin{equation} \mathcal{B}(\eta\to\pi^+\pi^-\mu^+\mu^-)<6\times10^{-7} \nonumber, \end{equation} \begin{equation} \mathcal{B}(\eta’(958)\to\pi^+\pi^-\mu^+\mu^-) = (2.80\pm0.14_{\text{(stat)}}\pm0.13_{\text{(syst)}}\pm0.14_{\phi})\times10^{-5} \nonumber, \end{equation} where the first uncertainty is statistical, the second is systematic, and the third is the uncertainty associated with the branching fraction of the $\phi\to\mu^+ \mu^-$ decay which is used for normalization. This marks the first-ever experimental search for $\eta’(958)\to\mu^+\mu^+$, and the most precise value of $\mathcal{B}(\eta\to\mu^+\mu^-)$ to date. INTNOTELHCBCONFPUBL