  In this paper we prove that there are finitely many modular curves that admit
a smooth plane model. Moreover, if the degree of the model is greater than or
equal to 19, no such curve exists. For modular curves of Shimura type we show
that none can admit a smooth plane model of degree 5, 6 or 7. Further, if a
modular curve of Shimura type admits a smooth plane model of degree 8 we show
that it must be a twist of one of four curves.
