We compare a number of GARCH and stochastic volatility (SV) models using nine series of oil, petroleum product and natural gas prices in a formal Bayesian model comparison exercise. The competing models include the standard models of GARCH(1,1) and SV with an AR(1) log-volatility process and more flexible models with jumps, volatility in mean and moving average innovations. We find that: (1) SV models generally compare favorably to their GARCH counterparts; (2) the jump component substantially improves the performance of the standard GARCH, but is unimportant for the SV model; (3) the volatility feedback channel seems to be superfluous; and (4) the moving average component markedly improves the fit of both GARCH and SV models. Overall, the SV model with moving average innovations is the best model for all nine series.