Onset of droplet motion is important for various applications, including dropwise condensation and water management in fuel cells. In order to determine critical conditions for onset of motion and specific motion mode, a general problem having a droplet sheared by a gas stream on inclined surface is investigated. Two criterion equations are theoretically established for onset of droplet sliding and rolling independently, including dimensionless parameters of Bond number (Bn), Ohnesorge number (Oh) and Weber number (We), inclination angle parameter and wettability parameters. The criterion equations predict critical gas velocity, maximum droplet radius, and "sliding angle" or "rolling angle". Droplet sliding predictions agree well with experimental data in the literature. Criterion surfaces of sliding or rolling are constructed to verify if sliding or rolling can be initiated, influenced by both equilibrium contact angle and contact angle hysteresis. By coupling the criterion equations of sliding and rolling, we develop a mode selection criterion equation, which is only dependent on equilibrium contact angle Be. Three regions are clarified: (1) for 126.3 degrees < theta(e) < 147.0 degrees, a droplet rolls if Bn < Bn-t and slides if Bn > Bn-t, where Bnt is the transition Bond number; (2) for theta(e) < 126.3 degrees a droplet only slides; (3) for theta(e) > 147.0 degrees, a droplet only rolls. The theoretically determined 147.0 degrees contact angle is newly recommended as the contact angle boundary between hydrophobicity and super-hydrophobicity. One of the applications of this work is to provide a general guideline for droplet detachment or retention in fuel cells and condensers. (C) 2018 Elsevier Ltd. All rights reserved.