The aim of this paper is to give an uniform approach to different kinds of degenerate hyperbolic Cauchy problems. We prove that a weakly hyperbolic equation, satisfying an intermediate condition between effective hyperbolicity and the $C^{infty}$ Levi condition, and a strictly hyperbolic equation with non-regular coefficients with respect to the time variable can be reduced to first order systems of the same type. For such a kind of systems, we prove an energy estimate in Sobolev spaces (with a loss of derivatives) which gives the well-posedness of the Cauchy problem in $C^{infty}$. In the strictly hyperbolic case, we also construct the fundamental solution and we describe the propagation of the space singularities of the solution which is influenced by the non-regularity of the coefficients with respect to the time variable.