Lamellar two-phase composites that result from either solid-liquid or solid-solid growth fronts are in general not perfectly periodic. Rather, they exhibit irregularities that can arise either from the presence of defects (grain boundaries or foreign-phase particles) or from intrinsic dynamical instabilities of the growth front. Many instabilities corresponding to different bifurcation types have been found and characterized in lamellar growth of eutectic alloys, both in experiments and simulations. Some parallels of these instabilities, such as lamellar-to-rod transitions or tip splitting of the growing lamellae, have also been observed during experimental investigations of discontinuous precipitation reactions. Using phase-field simulations, we investigate here the morphological instabilities of growing lamellar precipitates during discontinuous precipitation. The stability of the lamellae depends on their spacing ? as well as on the dimensionless supersaturation ?. The instabilities set the upper and lower limits of the range of the stable periodic lamellae at fixed ?, as well as an upper limit for ? at a fixed spacing ?. In order to assess the importance of surface diffusion along interfaces and grain boundaries for the onset of the instabilities, we have carried out simulations for two different sets of parameters. In both, bulk diffusion is present, but in the first one, we ignore the presence of interfaces by setting all surface diffusivities equal to zero, whereas in the second one we allow for surface diffusion. We have observed that the presence of surface diffusion shifts the onset of both the oscillatory and tip splitting instabilities towards higher values of ?, which results in slightly larger stability range for the growing lamellae. Stability curves are presented for the latter by plotting the variation of the lamellar spacing ? as a function of the scaled supersaturation ?.