We study a very general class of first-order linear hyperbolic systems that both become weakly hyperbolic and contain singular lower-order coefficients at a single time t = 0. In "critical" weakly hyperbolic settings, it is well-known that solutions lose a finite amount of regularity at t = 0. Here, we both improve upon the analysis in the weakly hyperbolic setting, and we extend this analysis to systems containing critically singular coefficients, which may also exhibit modified asymptotics and regularity loss at t = 0. In particular, we give precise quantifications for (1) the asymptotics of solutions as t approaches 0, (2) the scattering problem of solving the system with asymptotic data at t = 0, and (3) the loss of regularity due to the degeneracies at t = 0. Finally, we discuss a wide range of applications for these results, including weakly hyperbolic wave equations (and equations of higher order), as well as equations arising from relativity and cosmology (e.g. at big bang singularities). This is joint work with Bolys Sabitbek.