Constant pressure Monte Carlo simulations of a coarse-grained off-lattice model for monolayers of amphiphilic molecules at the air-water interface are presented. Our study focuses on phase transitions within a monolayer rather than on self-aggregation. We thus model the molecules as stiff chains of Lennard-Jones spheres with one slightly larger repulsive end bead (head) grafted to a planar surface. Depending on the size of the head, the temperature and the pressure, we find a variety of phases, which differ in tilt order (including tilt direction), and in positional order. In particular, we observe a modulated phase with a striped superstructure. The modulation results from the competition between two length scales, the head size, and the tail diameter. As this mechanism is fairly general, it may conceivably also be relevant in experimental monolayers. We argue that the superstructure would be very difficult to detect in a scattering experiment, which perhaps accounts for the fact that it has not been reported so far. Finally the effect of varying the chain length on the phase diagram is discussed. Except at high pressures and temperatures, the phase boundaries in systems with longer chains are shifted to higher temperatures.