  We describe spectral data for singular fibres of the
$\mathsf{SL}(2,\mathbb{C})$-Hitchin fibration with irreducible and reduced
spectral curve. Using Hecke transformations we give a stratification of these
singular spaces by fibre bundles over Prym varieties. By analysing the
parameter spaces of Hecke transformations this describes the singular Hitchin
fibres as compactifications of abelian group bundles over abelian torsors. We
prove that a large class of singular fibres are themselves fibre bundles over
Prym varieties. As applications we study irreducible components of singular
Hitchin fibres and give a description of $\mathsf{SL}(2,\mathbb{R})$-Higgs
bundles in terms of these semi-abelian spectral data.
