We investigate neutron stars in scalar-tensor theories. We examine their secular stability against spherically symmetric perturbations by use of a turning point method. For some choices of the coupling function contained in the theories, the number of the stable equilibrium solutions changes and the realized equilibrium solution may change discontinuously as the asymptotic value of the scalar field or total baryon number is changed continuously. The behavior of the stable equilibrium solutions is explained by fold and cusp catastrophes. Whether or not the cusp catastrophe appears depends on the choice of the coupling function. These types of catastrophes are structurally stable. Recently discovered spontaneous scalarization, which is a nonperturbative strong-field phenomenon due to the presence of the gravitational scalar field, is well described in terms of the cusp catastrophe.