Empirical estimates of CRRA and CARA utility
There are many estimates in the literature. For example, Havranek (2013) does a meta-analysis of avalible results and argues for a value of intertemporal elasticity (inverse of sigma in your notation) around 0.3-0.4. But it might also depend on what your goal is - the single parameter in CRRA utility controls both risk aversion and intertemporal smoothing motive, so a calibration for asset-pricing model might need to differ from let's say a deterministic growth model.