  A search is performed for rare and forbidden charm decays of the form
$D_{(s)}^+ \to h^\pm \ell^+ \ell^{(\prime)\mp}$, where $h^\pm$ is a pion or
kaon and $\ell^{(')\pm}$ is an electron or muon. The measurements are performed
using proton-proton collision data, corresponding to an integrated luminosity
of $1.6\text{fb}^{-1}$, collected by the LHCb experiment in 2016. No evidence
is observed for the 25 decay modes that are investigated and $90\%$ confidence
level limits on the branching fractions are set between $1.4\times10^{-8}$ and
$6.4\times10^{-6}$. In most cases, these results represent an improvement on
existing limits by one to two orders of magnitude.
