In this talk I will survey some recent results about the solvability of operators with multiple characteristics. It will be seen in the first place that, although degenerate, with some operators it is possible to associate a dynamics which describes the propagation of singularities/regularity, and hence are semi-globally solvable. I will next show another class of operators, modelled after the Kannai operator, with a possibly very degenerate characteristic set, which are still locally solvable in $L^2$.