Keywords: conformal, metric, shape, curves, harmonic, multiscale Abstract: We will present several families of metrics on plane curves, each of which is based on some multi-scale representation and is equivalent to a Sobolev-type norm. These metrics arise when trying to characterize local regularity of functions and curves. The underlying techniques are borrowed from harmonic and complex analysis. We will present theoretical and practical results; in particular we will show experimental results on the MPEG-7 Shape 1b test dataset.