A numerical study of the temporal evolution of three-dimensional instabilities on a planar liquid sheet/jet segment is performed using Direct Numerical Simulation and level-set analysis. Use is made of an unsteady 3D code with a finite-volume solver of the Navier-Stokes equations for the liquid stream and the adjacent gas, and a level-set method for the liquid/gas interface tracking. The initial profiles and the primary estimations are obtained from a 2D full-jet simulation. This study reveals insights into the development of three-dimensional instabilities on liquid sheets, which result in formation of lobes, holes, bridges, ligaments, and eventually droplets. Vortex dynamics is used to examine the mechanisms responsible for the distortion of the liquid sheet surface from primary Kelvin-Helmholtz perturbations into three-dimensional waves. Various breakup patterns occur at different flow regimes. The gas-to-liquid density and viscosity ratios, the Reynolds (Re) and Weber (We) numbers, and geometrical parameters such as the liquid sheet thickness to initial wavelength ratio are the most important parameters. At low/medium Re, hairpin vortices form on the braid and overlap with the lobe hairpins, causing thinning of the lobe sheets and resulting in formation of holes and bridges, which later break into ligaments and droplets. At higher Re, perforation of the lobes is hindered due to splitting of the original vortex rollers. The advection of the fast-moving eddy causes corrugations at high Re. The ligaments are then formed due to stretching of these corrugations. More ligaments with thinner cross-section are formed at high Re. The Weber number also has significant effect on the size of the ligaments and droplets. High surface tension prevents the thinning of the lobe sheet and precludes its perforation. The anti-symmetric mode is dominant in the practical injector range of Re and We. For the thicker liquid sheet, the anti-symmetric waves appear later and their wavelength is larger than for the thin-sheet case. Vortex stretching and tilting contribute to the generation of streamwise vorticity for a wide range of Re. λ2 contours are used to relate the vorticity dynamics of the flow to the deformation of the liquid sheet surface and different stages of droplet formation, e.g. lobe formation and perforation, ligament formation, stretching and tearing.