The investigation of the spectral distances of graphs that started in [3] (I. Jovanovi\'c, Z. Stani\'c, Spectral distances of graphs, Linear Algebra Appl., 436 (2012) 1425--1435.) is continued by defining Laplacian and signless Laplacian spectral distances and considering their relations to the spectral distances based on the adjacency matrix of graph. Some separate results concerning the defined distances are given, and the initial spectral distances in certain sets of graphs are investigated. Computational data on Laplacian and signless Laplacian spectral distances are provided.