Interacting realization of cosmological singularities with variable vacuum energy

We examine an interacting dark matter--variable vacuum energy model for a spatially flat Friedmann-Roberston-Walker spacetime, focusing on the appearance of cosmological singularities such as \emph{big rip, big brake, big freeze}, and \emph{ big separation} along with abrupt events (\emph{infinite $\gamma$- singularity} and \emph{new w-singularity}) at late times. We introduce a phenomenological interaction which has a nonlinear dependence on the total energy density of the dark sector and its derivative, solve exactly the source equation for the model and find the energy density as function of the scale factor as well as the time dependence of the approximate scale factor in the neighborhood of the singularities. We describe the main characteristics of these singularities by exploring the type of interaction that makes them possible along with behavior of dark components near them. We apply the geometric Tipler and Kr\'olak method for determining the fate of time-like geodesic curves around the singularities. We also explore the strength of them by analyzing the leading term in some geometric invariants such as the square Riemann scalar and the Ricci scalar.