  We consider smooth flows preserving a smooth invariant measure, or,
equivalently, locally Hamiltonian flows on compact orientable surfaces and show
that, when the genus of the surface is two, almost every such locally
Hamiltonian flow with two non degenerate isomorphic saddle has singular
spectrum. More in general, singularity of the spectrum holds for special flows
over a full measure set of interval exchange transformations with a
hyperelliptic permutation (of any number of exchanged intervals), under a roof
with symmetric logarithmic singularities. The result is proved using a
criterion for singularity based on tightness of Birkhoff sums with exponential
tails decay. A key ingredient in the proof, which is of independent interest,
is a result on translation surfaces well approximated by single cylinders. We
show that for almost every translation surface in any connected component of
any stratum there exists a full measure set of directions which can be well
approximated by a single cylinder of area arbitrarily close to one. The result,
in the special case of the stratum $\mathcal{H}(1,1)$, yields rigidity sets
needed for the singularity result.
