The Kasner spacetimes are solutions to Einstein’s equations that play a decisive role in the study of so-called spacelike singularities in GR. In this talk, we study two model hyperbolic systems in Kasner spacetimes, namely the scalar wave equation and the linearized Einstein(-scalar field) system, where one can precisely quantify blow-up asymptotics near the t = 0 singularity of Kasner. We present recent results deriving a sharp correspondence between Cauchy data and asymptotic data at the singularity, which reveals interesting phenomena regarding gains and losses of regularity.