Let $(M,\omega)$, be presymplectic, then can we say, we have a foliation of $M$, with tangent spaces $ker\omega$.What can we say about its trajectories. ?

If your definition of presymplectic involves constant rank and $\ker\omega$ defines a smooth distribution of hyperplanes in the tangent bundle, then an application of the Frobenious theorem shows that you have a foliation tangent to $\ker \omega$. I'm not sure that one can say more than that in general. 

