This paper is devoted to some singular ODEs flows X(•, x) for x in the d-dimensional torus T d , solutions to ∂ t X(t, x) = b(X(t, x)), with the initial condition X(0, x) = x. On the one hand, in dimension two assuming that the Herman rotation set for the flow X is parallel to some integer vector k = 0 R 2 , and that the component b(x) • k ⊥ is dominated by a function only depending on x • k ⊥ and changing sign, we prove that the flow X cannot have any invariant probability measure with positive Lebesgue density. As a by-product such a flow is also devoid of any first integral with periodic gradient. This result is illustrated by an explicit class of singular flows which contains in particular two-dimensional Euler flows. We also extend this class in dimension three restricting ourselves to the singular invariant measures. On the other hand, we consider a vector field b satisfying the non-negativity condition b • (∇u-∇u) ≥ 0 in T d but not identically null, for some periodic gradient field ∇u. Then, the flow associated with b turns out to have only singular singular invariant probability measures in any dimension d ≥ 2, and to be not integrable in dimension two. This class is illustrated by rather general examples.