Complete bipartite graphs flexible in the plane

A complete bipartite graph $K_{3,3}$, considered as a planar linkage with joints at the vertices and with rods as edges, is in general inflexible, that is, it admits only motions as a whole. Two types of its paradoxical mobility were found by Dixon in 1899. Later on, in a series of papers by several different authors the question of the flexibility of $K_{m,n}$ was solved for almost all pairs $(m,n)$. We solve it for all complete bipartite graphs in the Euclidean plane, as well as on the sphere and hyperbolic plane. We give independent self-contained proofs without extensive computations, which are almost the same in the Euclidean, hyperbolic and spherical cases.