Three computer algorithms were used to investigate the size distributions of interstices in random packings of spheres whose size distributions obeyed the log-normal distribution. The structure of the interstices was modelled in terms of a network of large pores interconnected by channels and the effects of the mode of packing and the standard deviation of the spheres on this network were measured. As the standard deviation of the particle sizes increased, the sizes of the interstices were transformed from a narrow, centred distribution to one which was broader and less ordered. This transformation was associated with an increase in the mean size of the pores, a decrease in their number and a transition from octahedral to tetrahedral pore shapes. In general, these effects were more pronounced in random loose packings than in random close packings. The interstice channels were measured by percolating spheres of different radii through the lattice. The range of radii which were able to successfully percolate the packing was found to increase with the standard deviation of lattice particle sizes whereas the number of distinct percolation paths decreased.