When to do studentization? I see texts on using studentized residuals when discussing problems in regression. Are there rules of thumb for when to apply studentized residuals? I mean, when can we tell normal residue plot is insufficient to tell if there are outliers and high leverage points? So far I see, normal residual plots seem sufficient.  
 A: Studentisation is the process of standardising observations using the sample mean and variance.  This process is generally necessary to obtain ancillary statistics for the purposes of testing a model assumption in cases where there are unknown parameters.  When you are testing to see if data come from some posited distribution, you are usually actually testing a family of distributions that are indexed by a nuisance parameter.  Studentisation helps you obtain observable values that have a fixed distribution under all the particular distributions in that family (i.e., ancillary statistics).
Testing the normal distribution using studentised values: Suppose you observe a sample of values $X_1, ..., X_n$ drawn from an exchangeable series of observable values (i.e., they are IID random variables) and you want to test whether these values follow a normal distribution.
You can do this by plotting a QQ-plot, comparing the sample quantiles to the theoretical quantiles of the posited distribution.  The difficulty with this is that "the normal distribution" is actually a family of distributions that depend on an unknown mean parameter $\mu$ and standard deviation parameter $\sigma$.  The theoretical quantiles of "the normal distribution" depend on these two parameters, so what are the theoretical quantiles you are supposed to be plotting?
To deal with this issue, you can studentise your data to obtain values that are (marginal) ancillary statistics.  Using the sample mean $\bar{X}$ and sample standard deviation $S$, you can form the statistics:
$$T_i \equiv \sqrt{\frac{n}{n-1}} \cdot \frac{X_i - \bar{X}}{S}.$$
These are the studentised values corresponding to the original observable values.  (The term at the front is to adjust for the variance suppression that comes from using the sample mean for the central location.)  Now, if $X_1, ..., X_n \sim \text{IID N} (\mu, \sigma^2)$ then it can be shown that:$\dagger$ 
$$T_1, ..., T_n \sim \text{Student's } T(\text{df} = n-1).$$
Notice that the distribution of these quantities does not depend on the parameters $\mu$ and $\sigma$.  This means that the studentised values take on a fixed distribution, so long as the underlying observable values take on any normal distribution.  Hence, you can test if the underlying values are normal by comparing the studentised quantities to a Student's T distribution.  This can be done by forming a QQ-plot, where you can now calculate the exact theoretical quantiles, owing to the fact that these do not depend on the parameters of the normal distribution.

$\dagger$ Note: The studentised values are not quite IID in this case.  There is some correlation between the values owing to estimation of the mean and variance.  For large $n$ the studentised values become asymptotically independent.
A: The advantage of studentized residuals is that they have a known (t with N - k - 1 df) distribution, hence you can do significance tests of the residuals. You can't do that with regular residuals (or standardized residuals).
If you want to do visual inspection to look for outliers and leverage points, regular residuals are fine. 
