We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces and show that almost every such locally Hamiltonian flow with only simple saddles has singular spectrum. Furthermore, we prove that almost every pair of such flows is spectrally disjoint. More in general, singularity of the spectrum and pairwise disjointness holds for special flows over a full measure set of interval exchange transformations under a roof with symmetric logarithmic singularities. The spectral result is proved using a criterion for singularity based on tightness of Birkhoff sums with exponential tails decay and the cancellations proved by the last author to prove absence of mixing in this class of flows, by showing that the latter can be combined with rigidity. Disjointness of pairs then follows by producing mixing times (for the second flow), using a new mechanism for shearing based on resonant rigidity times.
