Unsteady gravity-driven flow of a thin slender rivulet of a non-Newtonian power-law fluid on a plane inclined at an angle a to the horizontal is considered. Unsteady similarity solutions are obtained for both converging sessile rivulets (when 0 < alpha < pi/2) in the case x < 0 with t < 0, and diverging pendent rivulets (when pi/2 < alpha < pi) in the case x > 0 with t > 0, where x denotes a coordinate measured clown the plane and t denotes time. Numerical and asymptotic methods are used to show that for each value of the power-law index N there are two physically realisable solutions, with cross-sectional profiles that are 'single-humped' and 'double-humped', respectively. Each solution predicts that at any time t the rivulet widens or narrows according to vertical bar x vertical bar((2N+1)/2(N+1)) and thickens or thins according to vertical bar x vertical bar(N/(N+1)) as it flows down the plane: moreover, at any station x. it widens or narrows according to vertical bar t vertical bar(-N/2(N+1)) and thickens or thins according to vertical bar t vertical bar(-N/(N+1)). The length of a truncated rivulet of fixed volume is found to behave according to vertical bar t vertical bar(N/(2N+1)). (c) 2010 Elsevier B.V. All rights reserved.