  For graphs $G$ and $H$, let $G {\displaystyle\smash{\begin{subarray}{c}
\hbox{$\tiny\rm rb$} \\ \longrightarrow \\ \hbox{$\tiny\rm p$}
\end{subarray}}}H$ denote the property that for every proper edge-colouring of
$G$ there is a rainbow $H$ in $G$. It is known that, for every graph $H$, an
asymptotic upper bound for the threshold function $p^{\rm rb}_H=p^{\rm
rb}_H(n)$ of this property for the random graph $G(n,p)$ is
$n^{-1/m^{(2)}(H)}$, where $m^{(2)}(H)$ denotes the so-called maximum
$2$-density of $H$. Extending a result of Nenadov, Person, \v{S}kori\'c, and
Steger [J. Combin. Theory Ser. B 124 (2017),1-38] we prove a matching lower
bound for $p^{\rm rb}_{K_k}$ for $k\geq 5$. Furthermore, we show that $p^{\rm
rb}_{K_4} = n^{-7/15}$.
