Kaimanovich and Masur showed that a random walk on the mapping class group for an initial distribution with finite first moment and whose support generates a nonelementary subgroup, converges almost surely to a point in the space P MF of projective measured foliations on the surface. This defines a harmonic measure on P MF. Here, we show that when the initial distribution has finite support, the corresponding harmonic measure is singular with respect to the natural Lebesgue measure on P MF.