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The shapelets method for image analysis is based upon the decomposition of localised objects into a series of orthogonal components with 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. These frequently provide a more compact parameterisation, and can be interpreted in an intuitive way. Image manipulation in shapelet space is simplified by the concise expressions for linear coordinate transformations; 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 upon 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 observational 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 at http://www.astro.caltech.edu/~rjm/shapelets/ ., 14 pages, 11 figures. Corrected minor errata, including equation (32). Accompanying IDL software package is available from the shapelets web site at http://www.astro.caltech.edu/~rjm/shapelets/