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We report the results of a computer investigation of sets of mutually orthogonal Latin squares (MOLS) of small order. For n 9 we: (1) determine the number of orthogonal mates for each species of Latin square of order n; (2) calculate the proportion of Latin squares of order n that have an orthogonal mate, and the expected number of mates when a square is chosen uniformly at random; (3) classify all sets of MOLS of order n up to various different notions of equivalence. We also provide a tripledoi:10.1090/mcom/3010 fatcat:ougyaxarqzeujihucyyk54eb3y