Curves on spline surfaces such as NURBS have a long tradition in 3-D modeling, for instance as trimming curves or as motion paths. Now, as subdivision surfaces have become very popular, we seek a method for constructing spline-like curves on subdivision surfaces, beginning with a study of the Loop scheme. By using the natural parametrization of these surfaces, we find an appropriate construction based on subdivision leading to B-spline curves. With the help of typical graphics hardware, these curves can be displayed and edited in real time on a fine approximation to the limit surface.