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Non-conformal supercurrents in six dimensions are described, which contain the trace of the energy-momentum tensor and the gamma-trace of the supersymmetry current amongst their component fields. Within the superconformal approach to N=(1,0) supergravity, we present various distinct non-conformal supercurrents, one of which is associated with an O(2) (or linear) multiplet compensator, while another with a tensor multiplet compensator. We also derive an infinite class of non-conformal supercurrents involving O(n) multiplets with n > 2. As an illustrative example we construct the relaxed hypermultiplet in supergravity. Finally, we put forward a non-conformal supercurrent in the N= (2 0) supersymmetric case.