An aircraft wake is made of counter-rotating vortices and is known to be affected by a long (Crow) and a short (elliptic) wavelength instabilities. Numerical investigations on the three-dimensionnal instabilities and transient growth of such dipole are performed. By means of a three-dimensionnal linear stability analysis, we retrieve the instability bands corresponding to the Crow and elliptic modes but we also observe less unstable oscillatory modes with very broad peaks. The transient growth of perturbations on this dipole, investigated by computing the optimal linear perturbations with a direct-adjoint technique, demonstrates the crucial role of the region of maximal strain at short time and of the hyperbolic point at intermediate time. Investigations on the three-dimensionnal dynamics of trailing vortices in stratified fluids are performed. The elliptic instability is almost unaffected by weak and moderate stratifications.