Abstract: We calculate the density matrix for the decay of a polarised top quark into a polarised W boson and a massive 17 quark, for the most general Wth vertex arising from dimension-six gauge-invariant effective operators. We show that, in addition to the well-known W helicity fractions, for polarised top decays it is worth defining and studying the transverse and normal W polarisation fractions, that is, the W polarisation alone two directions orthogonal to its momentum. In particular, a rather simple forward-backward asymmetry in the normal direction is found to be very sensitive to complex phases in one of the Wth anomalous couplings. This asymmetry, which indicates a normal W polarisation, can be generated for example by a P-odd. T-odd transition electric dipole moment. We also investigate the angular distribution of decay products in the top quark rest frame, calculating the spin analysing powers for a general Wth vertex. Finally we show that, using a combined fit to top decay observables and the t W cross section, at LHC it will be possible to obtain model-independent measurements of all the (complex) Wth couplings as well as the single top polarisation. Implications for spin correlations in top pair production are also discussed.