We study the dynamics of instability and pattern formation in a slender elastic thread that is continuously fed onto a surface moving at constant speed V in its own plane. As V is decreased below a critical value V(c), the steady "dragged catenary" configuration of the thread becomes unstable to sinusoidal meanders and thence to a variety of more complex patterns including biperiodic meanders, figures of 8, "W," "two-by-one," and "two-by-two" patterns, and double coiling. Laboratory experiments are performed to determine the phase diagram of these patterns as a function of V, the thread feeding speed U, and the fall height H. The meandering state is quantified by measuring its amplitude and frequency as functions of V, which are consistent with a Hopf bifurcation. We formulate a numerical model for a slender elastic thread that predicts well the observed steady shapes but fails to predict the frequency of the onset of meandering, probably because of slippage of the thread relative to the belt. A comparison of our phase diagram with the analogous diagram for a thread of viscous fluid falling on a moving surface reveals many similarities, but each contains several patterns that are not found in the other.