Surfactants at air/water interfaces are often subjected to mechanical stresses as the interfaces they occupy are reduced in area. The most well characterized forms of stress relaxation in these systems are first order phase transitions from lower density to higher density phases. Here we study stress relaxation in lipid monolayers that occurs once chemical phase transitions have been exhausted. At these highly compressed states, the monolayer undergoes global mechanical relaxations termed collapse. By studying four different types of monolayers, we determine that collapse modes are most closely linked to in-plane rigidity. We characterize the rigidity of the monolayer by analyzing in-plane morphology on numerous length scales. More rigid monolayers collapse out-of-plane via a hard elastic mode similar to an elastic membrane, while softer monolayers relax in-plane by shearing.