When to use skew normal regression via MCMC (mixture models)? When do I use skew normal or skew t regression via MCMC? Do I use them when the data are heavily skewed, for example income data? Or do I fit a normal regression model first and inspect the residuals of the normal regression and if the residuals are skewed, then apply a skew normal regression?
 A: 
when do i use skew normal or skew t Regression via MCMC? 

If in that question you're just interested in the circumstances in which you'd use a skew-normal or a skew-t model, why specify the method of estimation (MCMC)? In respect of when to use it, why would it matter whether I used MLE or MCMC or methods of moments?

Do I use them when the data are heavily skewed, for example income data? 

It depends on which skew-normal and which skew-t you mean.
If you mean this skew normal distribution, the skew normal isn't nearly skewed enough for typical income data. 
In some circumstances, you might use it for the log of income data.
This skew-t seems to be used for somewhat more skewed situations (it's not 100% clear that it's the same object you intend though - you should always give enough details for people to be certain of exactly the distribution you mean).
Azzalini's skew-t seems to be the above skew-normal divided by the square root of (an independent chi-square on its df). I don't think its the same distribution as the skew-t as in that paper I link above, though it, too, seems as if it is able to be more skew than the skew-normal. Azzalini$^{[1]}$ seems to have used the log-skew-t for income, suggesting that in general that this skew-t may itself not be skewed enough for incomes.

Or do I fit a normal Regression model first and inspect the residuals of the normal Regression and if the residuals are skewed, then I apply a skew normal Regression?

A normal distribution almost certainly won't be suitable as a model for incomes.
[1]: A. Azzalini, T. Dal_Cappello and S. Kotz (2003). Log-skew-normal and log-skew-t distributions as models for family income data. J. Income Distribution, 11, no. 3-4, 12-20.
