Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schr ̈odinger type equations involving a fractional Laplacian in the one-dimensional case. By an appropriate choice of the dis- persive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up ver- sus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states, and the long time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schr ̈odinger