We present results of numerical simulations of local-optimal trajectories of a space probe with a flat solar sail which moves from a circular Earth orbit to near-Sun regions. We examine the planar (ecliptic) and three-dimensional transfer cases. We consider the upper limit of the sail temperature as the basic restriction. The local-optimal control law of the sail orientation angle was chosen in such a way that the rate of decrease of the orbital focal parameter reaches its maximal value. In the three-dimensional case, the requirement of the maximization of the orbital plane rotation rate was added. We also investigated complex control modes of the sail orientation for near-Sun orbits and of some `falling onto the Sun' trajectories. These computer simulations allow us to examine the durations of some sail missions and to investigate the evolution of osculating elliptical orbits.