AFM micro- or nanoindentation is a powerful technique for mapping the elasticity of materials at high resolution. When applied to soft matter, however, its accuracy is equivocal. The sources of the uncertainty can be methodological or analytical in nature. In this paper, we address the lack of practicable nonlinear elastic contact models, which frequently compels the use of Hertzian models in analyzing force curves. We derive and compare approximate force-indentation relations based on a number of hyperelastic general strain energy functions. These models were applied to existing data from the spherical indentation of native mouse cartilage tissue as well as chemically crosslinked poly(vinyl alcohol) gels. For the biological tissue, the Fung and single-term Ogden models were found to provide the best fit of the data while the Mooney-Rivlin and van der Waals models were most suitable for the synthetic gels. The other models (neo-Hookean, two-term reduced polynomial, Fung, van der Waals, and Hertz) were effective to varying degrees. The Hertz model proved to be acceptable for the synthetic gels at small strains (<20% for the samples tested). Although this finding supports the generally accepted view that many soft elastic materials can be assumed to be linear elastic at small strains, we propose the use of the nonlinear models when evaluating the large-strain indentation response of gels and tissues.