The evolution of Laguerre-Gaussian (LG) beams in the fractional Schrödinger equation (FSE) with Gaussian noise disturbance is numerically investigated. Without noise disturbance, the peak intensity of LG beams increases with the increment of radial or azimuthal indices, and the turning point of the peak intensity between different radial indices exists. As propagation distance gets longer, the intensity of the outermost sub-lobe exceeds that of the main lobe. When Gaussian noise is added, for a given noise level, the stability of peak intensity is enhanced as the Lévy index increases, while the center of gravity shows the opposite phenomenon. Moreover, the increment of the radial index can weaken the stability of the center of gravity. We also investigate the stability of the peak intensity of Airy beams in the FSE, and generally, the stability of LG beams is better than that of Airy beams. All these properties show that LG beams modeled by the FSE have potential applications in optical manipulation and communications.