The conditions are derived for the onset of lateral instabilities in planar waves propagating in a chemical system based on cubic autocatalysis coupled with autocatalyst decay, represented by the parameter kappa. The results from a linear stability analysis of the corresponding traveling wave equations are presented and compared with predictions obtained from a "thin front" analysis. Instabilities arise when delta, the ratio of the diffusion coefficients of the reactant and autocatalyst, is greater than some critical value delta (crit), with delta (crit) being found to depend strongly on kappa. Numerical simulations of the full initial-value problem are also determined and confirm the theoretical predictions.