In concurrency theory, various semantic equivalences on labelled transition systems are based on traces enriched or decorated with some additional observations. They are generally referred to as decorated traces, and examples include ready, failure, trace and complete trace equivalence. Using the generalized powerset construction, recently introduced by a subset of the authors [13], we give a coalgebraic presentation of decorated trace semantics. This yields a uniform notion of canonical, minimal representatives for the various decorated trace equivalences, in terms of final Moore automata. As a consequence, proofs of decorated trace equivalence can be given by coinduction, using different types of (Moore-) bisimulation (up-to), which is helpful for automation.