Phase oscillator lattices subject to noise are one of the most fundamental systems in nonequilibrium physics. We have discovered a dynamical transition which has a significant impact on the synchronization dynamics in such lattices, as it leads to an explosive increase of the phase diffusion rate by orders of magnitude. Our analysis is based on the widely applicable Kuramoto-Sakaguchi model, with local couplings between oscillators. For one-dimensional lattices, we observe the universal evolution of the phase spread that is suggested by a connection to the theory of surface growth, as described by the Kardar-Parisi-Zhang (KPZ) model. Moreover, we are able to explain the dynamical transition both in one and two dimensions by connecting it to an apparent finite-time singularity in a related KPZ lattice model. Our findings have direct consequences for the frequency stability of coupled oscillator lattices.