  We study slow entropy in some classes of smooth mixing flows on surfaces. The
flows we study can be represented as special flows over irrational rotations
and under roof functions which are $C^2$ everywhere except one point
(singularity). If the singularity is logarithmic asymmetric (Arnol'd flows) we
show that in the scale $a_n(t)=n(logn)^t$ slow entropy equals 1 (the speed of
orbit growth is nlogn) for a.e. irrational $\alpha$. If the singularity is of
power type ($x^{-\gamma}$, $\gamma\in (0,1)$) (Kochergin flows) we show that in
the scale $a_n(t)=n^t$ slow entropy equals $1+\gamma$ for a.e. $\alpha$. We
show moreover that for local rank one flows slow entropy equals $0$ in the
scale $n(logn)^t$. As a consequence we get that a.e. Arnol'd and a.e. Kochergin
flow is never of local rank one.
