We examine a three-dimensional rivulet flowing down a vertical plane. There exists a basic state with fully developed, unidirectional flow and straight contact lines. In the absence of contact-angle hysteresis the slope of the contact angle versus contact-line speed relationship measures the mobility of these contact lines. The stability characteristics of flat rivulets subject to long wave disturbances are examined using lubrication theory. We find that kinematic-wave instabilities are predicted for wide rivulets or rivulets with rather immobile contact lines, while capillary break-up is predicted for narrower rivulets with mobile contact lines. We find for all cases that the expression for the growth rate depends weakly on slip between the liquid and solid near the contact line, but strongly on the shape of the rivulet and the mobility of the contact lines. We discuss in detail the mechanisms by which the contact lines affect the instabilities.