Popcorn disease of mulberry is caused by a fungus (Ciboria carunculoides). It occurs in late spring and early summer. The white mulberries are more susceptible to this disease. The disease manifests on the developing carpels and looks like popcorn kernels. It is a serious disease if the tree is being cropped for commercial purposes; however, it does no harm to the overall health of the trees, thus homeowners do not need to worry (if the tree is only used for ornamental or shade purposes). So, if this disease does occur what can be done to stop it? Sanitation is a good first step — clean up any infected material and remove it from the area where the tree is growing. There is very little else a homeowner can feasibly do to reduce the disease. Spraying the tree with Bordeaux mixture may help too, but getting coverage over the entire tree may be problematic. As with many fungal diseases, the severity will depend on the environmental conditions from year to year. Some years will be worse than others.
If you wish to read more on this particular disease, Texas A&M has a good, short fact sheet on it here.