Ken Moore (retired associative director of the North Carolina Botanical Gardens) calls the current state of mulberries a "taxonomic mess"... I couldn't agree more!
My main reason for creating this website is to provide some enlightenment to allow for an informed purchase choice. Although there are some nurseries that purposely mislead the public, most are just misinformed. My goal would be to have some uniformity in mulberry cultivars so that a cultivar from one nursery would be the identical cultivar from another nursery. Someday, in the not too distant future, genetic testing will be commonplace and inexpensive, and maybe then cultivars will be better identified. Several “scientific” (1,2) papers appearing on the internet claim to differentiate between the many species and cultivars by using a relatively quick and easy form of genetic testing called targeted sequencing. However, presenting this idea of targeted sequencing to unravel the “taxonomic mess” to a genetics expert, I received the following response: “This is a very complex group, yes, a taxonomic mess, that will not be solved by targeting sequencing. You require a much broader and comprehensive approach most likely via next generation sequencing where you can examine 100’s of nuclear and chloroplast genes if not the entire genome which should be available.”
There is an “old school” method which allows identification of one species of mulberry (but not the individual cultivars). Morus nigra can be identified with a high level of certainty by microscopic examination of their cell nucleus and nucleolus size. If you are interested in wanting to know if that mulberry tree you purchased as a Morus nigra is actually a Morus nigra, I will be glad to do a determination (at no charge). Email me and I will give you instructions on mailing a couple of leaves.
Morus nigra has a thick, substantial leaf, relative to most mulberry cultivars, and their leaf thickness can be used to help identify this favorite species:
Take a couple mature leaves (measuring at least 4 inches in length), and using sharp scissors, cut ten similar sized rectangular sections of leaf (do not include any portion of the primary vein). Stack them on top each other and hold them between your thumb and index finger with a bit of the stack protruding. With scissors against your fingertips, make a fresh cut through the stack, and squeezing gently, measure with a metric ruler. If the stack does not measure approximately four millimeters thick or more, then it most likely is not Morus nigra. It won't confirm that you have Morus nigra though because other cultivars also exist with leaves of similar thickness. By comparison, most cultivars of Morus alba and Morus rubra "stack up" to be two to three millimeters thick.
Measuring leaf thickness
Mulberry leaf shape and size are extremely variable between species, hybrids, and cultivars; even on the same plant, and often varying significantly according to the age of the plant or the amount of sunlight.
Typical Morus nigra leaf shape and size (1/2 inch mat squares)
The lower leaf surface of Morus rubra and Morus nigra is quite hairy, giving a soft pubescent feel when lightly stroked.
A distinguishing feature of Morus rubra is a pronounced acuminate leaf apex (tip of leaf).
Morus nigra (Noir de Spain) with lobed leaves
Morus rubra and some of its hybrids have an upper leaf surface that is highly textured.
Discovered and named in Japan,
Morus alba 'Itoguwa' grows into a small, non-fruiting shrub.
Morus alba cultivars will generally have an upper leaf surface that has a waxy, glossy appearance, while Morus rubra cultivars (and hybrids), and Morus nigra, will appear relatively dull.
Morus nigra and Morus rubra leaves (and most hybrids) will have a sandpaper like feel when stroked gently on the upper leaf surface in the direction of the petiole. Morus alba cultivars will feel nearly the same in either direction.
An excellent paper from Purdue University describing the difference between Morus rubra and Morus alba can be found here:
Morus alba: Waxed leaf appearance.
Morus rubra: Dull leaves with finer leaf margin serrations than alba.
The growth habits of various species and cultivars can also help in mulberry identification. Morus nigra has a gnarly, somewhat zigzagged growth appearance with relatively short internodes (distance between buds). Morus nigra has perhaps the slowest growth among the different Morus species (The Gerardi Dwarf also has similar growth habits, but is not a Morus nigra or Morus nigra hybrid).
Long, straight branches are a distinguishing feature of Morus alba cultivars; the exception being the contorted 'Unryu' and 'Tortuosa' cultivars.
Morus nigra; Note the large dark buds.
Morus alba; Long straight branches
Since most plants are bought in the winter months, devoid of foliage, bud size and color are useful in identifying mulberries. Morus nigra has large, dark brown or black buds (hence "nigra") relative to other mulberries.
An exemplary description between Morus rubra and Morus alba buds appears in the Purdue University "Red and White Mulberry in Indiana", as noted and linked above:
"Red mulberry has larger, more flattened buds that often sit off-center on the twig; the margin of each scale has a black band. White mulberry buds are smaller and more domed in the middle, and they usually sit along the center of the twig. The tip of each white mulberry bud is short and needle sharp, and each bud scale has a brown band which is not necessarily on the margin. There is usually a color difference in the twigs of the two species as well. Red mulberry twigs are very pale tan, while those of white mulberry tend to be pinkish-brown."
Large dark buds of Morus nigra
Morus alba: Small, light-colored buds
Buds of Morus rubra
Paper Mulberry (Broussonetia papyrifera)
Although not a true mulberry of the genus Morus, Paper Mulberry does look very similar to the Red Mulberry (Morus rubra). Its leaves are very rough on top, and exceedingly hairy on the underside (more so than Morus rubra). The discerning difference between them is Paper Mulberry has hair on its reddish leaf petioles and hair on its tender small branches.
Paper Mulberry does have a fruit that is edible but only grow this plant if you want IT to take over your yard and your neighbor's. Even if you only have one male plant (like Morus, they are usually male or female), new plants will spring up from underground roots anywhere and everywhere within fifty feet of the tree.
It is listed as an invasive plant in southeastern US states and over a dozen countries worldwide.
References (this page only):
(1) Genetic relatedness among cultivated and wild mulberry (Moraceae: Morus) as revealed by inter-simple sequence repeat analysis in China; Zhao, W., Zhou, Z., Miao, X., Wang, S., Zhang, L., Pan, Y. and Huang, Y. 
(2) Genetic diversity and relationships in mulberry (genus Morus) as revealed by RAPD and ISSR marker assays; Arvind K Awasthi, GM Nagaraja, GV Naik, Sriramana Kanginakudru, K Thangavelu and Javaregowda Nagaraju