Abstract A search is performed for heavy neutrinos in the decay of a W boson into two muons and a jet. The data set corresponds to an integrated luminosity of approximately $$3.0\, \text {fb} ^{-1} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>3.0</mml:mn> <mml:mspace/> <mml:msup> <mml:mtext>fb</mml:mtext> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> of proton–proton collision data at centre-of-mass energies of 7 and $$8\, \text {TeV} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>8</mml:mn> <mml:mspace/> <mml:mtext>TeV</mml:mtext> </mml:mrow> </mml:math> collected with the LHCb experiment. Both same-sign and opposite-sign muons in the final state are considered. Data are found to be consistent with the expected background. Upper limits on the coupling of a heavy neutrino with the Standard Model neutrino are set at $$95\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>95</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> confidence level in the heavy-neutrino mass range from 5 to $$50\, \text {GeV/}c^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>50</mml:mn> <mml:mspace/> <mml:mtext>GeV/</mml:mtext> <mml:msup> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> . These are of the order of $$10^{-3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:math> for lepton-number-conserving decays and of the order of $$10^{-4}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> </mml:math> for lepton-number-violating heavy-neutrino decays.