This looks like a similar question and didn't get many responses.
Omitting tests such as Cook's D, and just looking at residuals as a group, I am interested in how others use residuals when assessing goodness-of-fit. I use the raw residuals:
- in a QQ-plot, for assessing normality
- in a scatterplot of $y$ versus residuals, for eyeball check of (a) hetereoscedasticity and (b) serial autocorrelation.
For plotting $y$ versus residuals to examine the values for $y$ where outliers may occur, I prefer to use the studentized residuals. The reason for my preference is that it allows easy viewing of which residuals at which $y$-values are problematic, although standardised residuals provide an extremely similar result. My theory on which is used is that it depends on which university one went to.
Is this similar to how others use residuals? Do others use this number of graphs in combination with summary statistics?