With the help of the largest data samples of $J/psi$ and $psi(2S)$ events ever produced in $e^+e^-$ annihilations, the three singlet charmonium states, $eta_c(1S)$, $eta_c(2S)$ and $h_c(1P)$, have been extensively studied at the BESIII experiment. In this review, a survey on the most recent results, including a series of precision measurements and observations of their new decay modes, is presented, which indicates the further investigations on their decays are needed to understand their decay mechanisms and have precision tests of the theoretical models. At present, about eight times larger data samples of 10 billion $J/psi$ events and 3 billion $psi(3686)$ events were collected with the BESIII detector, and thus the prospects for the study of these three charmonium states is discussed extensively.
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