A bstract A search is performed for rare and forbidden charm decays of the form $$ {D}_{(s)}^{+}\to {h}^{\pm }{\mathrm{\ell}}^{+}{\mathrm{\ell}}^{\left(\prime \right)\mp } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mfenced> <mml:mi>s</mml:mi> </mml:mfenced> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>h</mml:mi> <mml:mo>±</mml:mo> </mml:msup> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mrow> <mml:mfenced> <mml:mo>′</mml:mo> </mml:mfenced> <mml:mo>∓</mml:mo> </mml:mrow> </mml:msup> </mml:math> , where h ± is a pion or kaon and ℓ (′)± is an electron or muon. The measurements are performed using proton-proton collision data, corresponding to an integrated luminosity of 1 . 6 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 × 10 − 8 and 6 . 4 × 10 − 6 . In most cases, these results represent an improvement on existing limits by one to two orders of magnitude.