The detection and identification of resonances is a key ingredient in the studies of stability of dynamical systems, with important applications for our Solar as well as for extrasolar systems. In this paper we study the detection of resonances and close encounters in the three-body problem using the fast Lyapunov indicator method. In order to investigate the close encounters, we needed to adapt the method to the model and its singularities. Our technical improvement lies in computation of the solutions of the singular variational equations by measuring the divergence of close initial conditions. We have used the Levi-Civita regularization for the integration of the equations of motion. As an application of the method, we show that it provides a correct detection of the tube manifolds related to the Lagrangian point L1 of the Sun–Jupiter system.