I'm somewhat surprised that no one has linked to this paper: Backus, Routledge, and Zin (2004) Exotic Preferences for Macroeconomists (this version has some fixed typos, vs the NBER print).
Their abstract is concise and extremely descriptive:
We provide a user's guide to 'exotic' preferences: nonlinear time aggregators, departures from expected utility, preferences over time with known and unknown probabilities, risk-sensitive and robust control, 'hyperbolic' discounting, and preferences over sets ('temptations'). We apply each to a number of classic problems in macroeconomics and finance, including consumption and saving, portfolio choice, asset pricing, and Pareto optimal allocations.
The paper itself does a great job of overviewing many options for "non-standard" preferences and utility. They introduce a preference, outline key features, and then apply them in a number of classic settings to give you a feel for what is going on.
If you are interested in using non-traditional preferences, you should certainly read this. I can't give extensive insight further than that -- I currently do nearly all my work with more traditional CRRA preferences.