The shapelets method for astronomical image analysis is based around the decomposition of localised objects into a series of orthogonal components with particularly convenient mathematical properties. We extend the “Cartesian shapelet” formalism from earlier work, and construct “polar shapelet” basis functions that separate an image into components with explicit rotational symmetries. This provides a more compact representation of typical galaxy shapes, and its physical interpretation is frequently more intuitive. Linear coordinate transformations can be simply expressed using this basis set, and shape measures (including object photometry, astrometry and galaxy morphology estimators) take a naturally elegant form. Particular attention is paid to the analysis of astronomical survey images, and we test shapelet techniques with real data from the Hubble Space Telescope. We present a practical method to automatically optimise the quality of an arbitrary shapelet decomposition in the presence of noise, pixellisation and a Point-Spread Function. A central component of this procedure is the adaptive choice of the shapelet expansion’s scale size and truncation order. A complete software package to perform shapelet image analysis is made available on the world-wide web.