Abstract We provide statistical analysis methods for samples of curves in two or more dimensions, where the image, but not the parameterization of the curves, is of interest and suitable alignment/registration is thus necessary. Examples are handwritten letters, movement paths, or object outlines. We focus in particular on the computation of (smooth) means and distances, allowing, for example, classification or clustering. Existing parameterization invariant analysis methods based on the elastic distance of the curves modulo parameterization, using the square-root-velocity framework, have limitations in common realistic settings where curves are irregularly and potentially sparsely observed. We propose using spline curves to model smooth or polygonal (Fréchet) means of open or closed curves with respect to the elastic distance and show identifiability of the spline model modulo parameterization. We further provide methods and algorithms to approximate the elastic distance for irregularly or sparsely observed curves, via interpreting them as polygons. We illustrate the usefulness of our methods on two datasets. The first application classifies irregularly sampled spirals drawn by Parkinson's patients and healthy controls, based on the elastic distance to a mean spiral curve computed using our approach. The second application clusters sparsely sampled GPS tracks based on the elastic distance and computes smooth cluster means to find new paths on the Tempelhof field in Berlin. All methods are implemented in the R-package “elasdics” and evaluated in simulations.