A search is performed for D-sJ(()()+) mesons in the reactions pp -> D(+KSX)-X-0 and pp -> D*(K+X)-K-0 using data collected at centre-of-mass energies of 7 and 8 TeV with the LHCb detector. For the DK-+(S)0 final state, the decays D(+) -> D-0 pi(+) with D-0 -> K-pi(+) and D-0 -> K-pi(+)pi(+)pi(-) are used. For DK-0(+), the decay D(0) -> D-0 pi(0) with D-0 -> K-pi(+) is used. A prominent D-s1(2536)(+) signal is observed in both DK-+(S)0 and DK-0(+) 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)(+) -> D*(+) K-S(0) is observed for the first time, at a significance of 6.9 sigma, and its branching fraction relative to the D*(s2)(2573)(+) -> (D+KS0) decay mode is measured.