The formation of two-dimensional Turing patterns in nonequilibrium chemical systems is studied by numerical simulations with an activator-substrate depletion model. The relative stabilities of the different hexagonal patterns and the striped patterns are discussed, in particular in the vicinity of the transition, and compared with the present state theory. The generic instabilities of the striped patterns are evidenced. In particular, we report the formation of stable zigzag patterns which share features both of one-dimensional and two-dimensional patterns. We also provide examples of temporal evolution in a weakly confined system.