  A complete set of linearly independent observables in Compton scattering with
arbitrarily polarised real photons off an arbitrarily polarised spin-1 target
is introduced, for the case that the final-state polarisations are not
measured. Adopted from the one widely used e.g. in deuteron photo-dissociation,
it consists of 18 terms: the unpolarised cross section, the beam asymmetry, 4
target asymmetries and 12 asymmetries in which both beam and target are
polarised. They are expressed by the helicity amplitudes and -- where available
-- related to observables discussed by other authors. As application to
deuteron Compton scattering, their dependence on the (isoscalar) scalar and
spin dipole polarisabilities of the nucleon is explored in Chiral Effective
Field Theory with dynamical Delta(1232) degrees of freedom at order
$e^2\delta^3$. Some asymmetries are sensitive to only one or two dipole
polarisabilities, making them particularly attractive for experimental studies.
At a photon energy of 100 MeV, a set of 5 observables is identified from which
one may be able to extract the spin polarisabilities of the nucleon. These are
experimentally realistic but challenging and mostly involve tensor-polarised
deuterons. Relative to Compton scattering from a nucleon, sensitivity to the
"mixed" spin polarisabilities $\gamma_{E1M2}$ and $\gamma_{M1E2}$ is increased
because of interference with the D wave component of the deuteron and with its
pion-exchange current. An interactive Mathematica 9.0 notebook with results for
all observables at photon energies up to 120 MeV is available from
<EMAIL_ADDRESS>  Note May 2017/April 2018: This arXiv version presents the article including
the corrections in the Errata Eur. Phys. J. A53 (2017) 113 and A54 (2018) 57 to
ibid. A49 (2013) 100. This affects Eqs. (2.12) to (2.14), Figs. 4, 7 to 10, 12,
14 to 16, 18 to 20, 22 and 23, and the corresponding, related text.
