A search is performed for $D^{(*)+}_{sJ}$ mesons in the reactions $pp \\to D^{*+} K^0_{\\rm S} X$ and $pp \\to D^{*0} K^+ X$ using data collected at centre-of-mass energies of 7 and 8 TeV with the LHCb detector. For the $D^{*+} K^0_{\\rm S}$ final state, the decays $D^{*+} \\to D^0 \\pi^+$ with $D^0 \\to K^- \\pi^+$ and $D^0 \\to K^- \\pi^+ \\pi^+ \\pi^-$ are used. For $D^{*0} K^+$, the decay $D^{*0} \\to D^0 \\pi^0$ with $D^0 \\to K^- \\pi^+$ is used. A prominent $D_{s1}(2536)^+$ signal is observed in both $D^{*+} K^0_{\\rm S}$ and $D^{*0} K^+$ final states. The resonances $D^*_{s1}(2700)^+$ and $D^*_{s3}(2860)^+$ are also observed, yielding information on their properties, including spin-parity assignments. The decay $D^*_{s2}(2573)^+ \\to D^{*+} K^0_{\\rm S}$ is observed for the first time, at a significance of 6.9 $\\sigma$, and its branching fraction relative to the $D^*_{s2}(2573)^+ \\to D^+ K^0_{\\rm S}$ decay mode is measured.
