  We study the spectral measures of conservative mixing flows on the 2-torus
having one degenerate singularity. We show that, for a sufficiently strong
singularity, the spectrum of these flows is typically Lebesgue with infinite
multiplicity. For this, we use two main ingredients: 1) a proof of absolute
continuity of the maximal spectral type for this class of non-uniformly
stretching flows that have an irregular decay of correlations, 2) a geometric
criterion that yields infinite Lebesgue multiplicity of the spectrum and that
is well adapted to rapidly mixing flows.
