  We investigate the dependence of polarisation observables in elastic deuteron
Compton scattering below the pion production threshold on the spin-independent
and spin-dependent iso-scalar dipole polarisabilities of the nucleon. The
calculation uses Chiral Effective Field Theory with dynamical Delta(1232)
degrees of freedom in the Small Scale Expansion at next-to-leading order.
Resummation of the NN intermediate rescattering states and including the Delta
induces sizeable effects. The analysis considers cross-sections and the
analysing power of linearly polarised photons on an unpolarised target, and
cross-section differences and asymmetries of linearly and circularly polarised
beams on a vector-polarised deuteron. An intuitive argument helps one to
identify kinematics in which one or several polarisabilities do not contribute.
Some double-polarised observables are only sensitive to linear combinations of
two of the spin-polarisabilities, simplifying a multipole-analysis of the data.
Spin-polarisabilities can be extracted at photon energies \gtrsim 100 MeV,
after measurements at lower energies of \lesssim 70 MeV provide high-accuracy
determinations of the spin-independent ones. An interactive Mathematica 7.0
notebook of our findings is available from<EMAIL_ADDRESS>