We implement the theory of resonant scattering in the context of Cosmic
Microwave Background (CMB) polarisation anisotropies. We compute the changes
in the E-mode polarisation (EE) and Temperature E-mode (TE) CMB power spectra
introduced by the scattering on a resonant transition with a given optical
depth $\tau_X$ and polarisation coefficient $E_1$. The latter parameter,
accounting for how anisotropic the scattering is, depends on the exchange of
angular momentum in the transition, enabling observational discrimination
between different resonances.
We use this formalism in two different scenarios: cosmological recombination
and cosmological reionisation. In the context of cosmological recombination,
we compute predictions in frequency and multipole space for the change in the
TE and EE power spectra introduced by scattering on the $H_{\alpha}$ and
$P_{\alpha}$ lines of hydrogen. This constitutes a fundamental test
of the standard model of recombination, and the sensitivity it
requires is comparable to that needed in measuring the primordial CMB B-mode
polarisation component. In the context of reionisation, we study the
scattering off metals and ions produced by the first stars, and find that
polarisation anisotropies, apart from providing a consistency test for
intensity measurements, give some insight on how reionisation
evolved. Since polarisation anisotropies have memory of how
anisotropic the line scattering is, they should be able to discern the OI
63.2$\mu$m transition from other possible transitions associated to OIII, NII,
NIII, etc. The amplitude of these signals are, however, between 10 to 100
times below the (already challenging) level of CMB B-mode polarisation
anisotropies.