This paper is devoted to some singular ODEs flows $X(\cdot,x)$ for $x$ in the $d$-dimensional torus~$\TT^d$, solutions to $\partial_t X(t,x)=b(X(t,x))$, 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 $k\neq 0_{\RR^2}$, and that the component $b(x)\cdot k^\perp$ is dominated by a function only depending on $x\cdot k^\perp$ and changing sign, we prove that the flow $X$ cannot have any invariant probability measure with positive Lebesgue density. As a by-product, if the vector field $b$ does not vanish in $\TT^2$, the flow $X$ is shown to be not integrable through a real-analytic first integral, {\em i.e.} it is devoid of any real-analytic first integral with a periodic gradient. This result is illustrated by an explicit class of singular flows which contains in particular some 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 $b\cdot(\nabla u-\overline{\nabla u})\geq 0$ in $\TT^d$ but not identically null, for some periodic gradient field~$\nabla u$. Then, the flow associated with $b$ turns out to have only singular singular invariant probability measures in any dimension $d\geq 2$. Moreover, the flow is not integrable in dimension two if $b$ does not vanish in~$\TT^2$. This class is illustrated by rather general examples.