A non-empty set G is said to be a groupoid relatively to a binary operation denoted by {·}, if for every ordered pair (a, b) of elements of G there is a unique element ab ∈ G. A groupoid (G, ·) is called a quasigroup if for every a, b ∈ G the equations a · x = b and y · a = b have unique solutions. We consider the quasigroups of properties (∗): left distributive, right distributive, left Bol, right Bol, left involutary, right involutary, left semimedial, right semimedial, left cancellative, right cancellative, non-associative loops, medial, paramedial, Ward, Cote and Manin quasigroups. We examine the following problem: Problem 1. How many non-isomorphic quasigroups of properties (∗) of order 3, 4, 5, 6, 7 do there exist? We have elaborated algorithms for generating and enumerating non-isomorphic quasigroups of small order of properties (∗). The results established here are related to the work in ([1,2,3,4,5]). Applying the algorithms elaborated, we prove the following results: