We construct counterexamples to the local existence of low-regularity solutions to elastic wave equations and to the ideal compressible magnetohydrodynamics (MHD) system in three spatial dimensions (3D). In particular, the Cauchy problems for 3D elastic waves and for MHD system are ill-posed in $H^3$ and $H^2$, respectively. This result generalizes Lindblad’s classic results on the scalar wave equation. Both elastic waves and MHD are physical systems with multiple wavespeeds. We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. Moreover, when the magnetic field is absent in MHD, we also provide a desired low-regularity ill-posedness result for the 3D compressible Euler equations, and it is sharp with respect to the regularity of the fluid velocity. Our proofs are based on a coalition of a carefully designed algebraic approach and a geometric approach. This talk is based on joint works with Xinliang An and Silu Yin.