Motivated by their important role in smooth dynamical systems, Lyapunov exponents have been conceived decades ago as a means to study the stability of cellular automata (CAs). More precisely, they quantify their sensitive dependence on initial conditions. As a next step towards the establishment of a dynamical systems theory of CAs that is inspired by its analogue for smooth dynamical systems, we introduce the concept of Lyapunov profiles of CAs. These constructs may be considered analogous to the Lyapunov spectra of higher-dimensional smooth dynamical systems. Doing so, we unify the competing approaches to Lyapunov exponents of CAs, as Lyapunov profiles capture both the spreading properties of a set of defects and the exponential accumulation rates of defects within this set.