This paper deals with nonholonomic systems in chained form with unknown covariance stochastic disturbances. The objective is to design the almost global adaptive asymptotical controllers in probability u0 and u1 for the systems by using discontinuous control. A switching control law u0 is designed to almost globally asymptotically stabilize the state x0 in both the singular x0 (t0)=0 case and the non-singular x0 (t0)≠0 case. Then the state scaling technique is introduced for the discontinuous feedback into the (x1, x2, …, xn)-subsystem. Thereby, by using backstepping technique the global adaptive asymptotical control law u1 has been presented for (x1, x2, …, xn) -subsystem for both different u0 in non-singular x0 (t0)≠0 case and the singular case x0 (t0)=0. The control algorithm validity is proved by simulation.