  We present an analysis of $13$ observables in Compton scattering on the
proton. Cross sections, asymmetries with polarised beam and/or targets, and
polarisation-transfer observables are investigated for energies up to the
$\Delta(1232)$ resonance to determine their sensitivity to the proton's dipole
scalar and spin polarisabilities. The Chiral Effective Field Theory Compton
amplitude we use is complete at N$^4$LO, $\mathcal{O(}e^2\delta^4)$, for photon
energies $\omega\sim m_\pi$, and so has an accuracy of a few per cent there. At
photon energies in the resonance region it is complete at NLO,
$\mathcal{O}(e^2\delta^0)$, and so its accuracy there is about $20$\%. We find
that for energies from pion-production threshold to about $250\;\mathrm{MeV}$,
multiple asymmetries have significant sensitivity to presently ill-determined
combinations of proton spin polarisabilities. We also argue that the broad
outcomes of this analysis will be replicated in complementary theoretical
approaches, e.g., dispersion relations. Finally, we show that below the
pion-production threshold, $6$ observables suffice to reconstruct the Compton
amplitude, and above it $11$ are required. Although not necessary for
polarisability extractions, this opens the possibility to perform "complete"
Compton-scattering experiments. An interactive Mathematica notebook, including
results for the neutron, is available from<EMAIL_ADDRESS>.
