We consider the problem of hypotheses testing with the basic simple hypothesis: observed sequence of points corresponds to stationary Poisson process with known intensity. The alternatives are either stationary self-exciting or stationary self-correcting point processes. In the first case we consider one-sided parametric and one-sided nonparametric alternatives and construct locally asymptotically uniformly most powerful tests. In the second case we consider one-sided parametric alternatives. The results of numerical simulations of the tests are presented.