The main goal of this paper is an application of Bayesian model comparison, based on the posterior probabilities and posterior odds ratios, in testing the explanatory power of the set of competing GARCH (Generalised Autoregressive Conditionally Heteroscedastic) specifications, all with asymmetric and heavy tailed conditional distributions. In building competing volatility models we consider, as an initial specification, GARCH process with conditional Student-t distribution with unknown degrees of freedom parameter, proposed by Bollerslev. By introducing skewness into Student-t family and by application the resulting class as a conditional distribution we generated various GARCH models, which compete in explaining possible asymmetry of both conditional and unconditional distribution of financial data. In order to make Student-t family skewed we consider various alternative methods recently proposed in the literature. In particular, we apply the hidden truncation mechanism, an approach based on the inverse scale factors in the positive and the negative orthant, order statistics concept, Beta distribution transformation and Bernstein density transformation. Additionally, we consider GARCH process with conditional a-Stable distribution. Based on the daily returns of hypothetical financial time series, we discuss the results of Bayesian comparison of alternative skewing mechanisms applied in the initial Student -t GARCH framework. We also check the sensitivity of model ranking with respect to structural changes in dynamics of considered time series. Additionally, we present formal Bayesian inference about conditional asymmetry of the distribution of the daily returns in all competing specifications on the basis of the skewness measure defined by Arnold and Groenveld.