We study in detail complex structures of homoclinic bifurcations in a three-dimensional rate-equation model of a semiconductor laser receiving optically injected light of amplitude K and frequency detuning w. Specifically, we find and follow in the (K,w)-plane curves of n-homoclinic bifurcations, where a saddle-focus is connected to itself at the n-th return to a neighborhood of the saddle. We reveal an intricate interplay of codimension-two double-homoclinic and T-point bifurcations. Furthermore, we study how these structures change with an additional parameter, the so-called linewidth enhancement factor alpha of the laser. In particular, we find folds (minima) of T-point bifurcation and double-homoclinic bifurcation curves, which are accumulated by infinitely many changes of the bifurcation diagram due to transitions through singularities of surfaces of homoclinic bifurcations. The injection laser emerges as a system that allows one to study codimension-two bifurcations of n-homoclinic orbits in a concrete vector field. At the same time, the bifurcation diagram in the (K,w)-plane is of physical relevance. An example is the identification of regions, and their dependence on the parameter alpha, of multi-pulse excitability where the laser reacts to a single small perturbation by sending out n pulses.