When should normalization never be used? Lately, there have been numerous questions about normalization
What are some of the situations where you never ever ever should normalize your data, and what are the alternatives?
 A: Of course one should never try to blindly normalize data if the data does not follow a (single) normal distribution.
For example one might want to rescale observables $X$ to all be normal with $(X-\mu)/\sigma$, but this can only work if the data is normal and if both $\mu$ and $\sigma$ are the same for all data points (e.g. $\sigma$ doesn't depend on $\mu$ in a particular $X$ range).
A: Whether one can normalize a non-normal data set depends on the application.  For example, data normalization is required for many statistical tests (i.e. calculating a z-score, t-score, etc.)  Some tests are more prone to failure when normalizing non-normal data, while some are more resistant ("robust" tests).  
One less-robust statistic is the mean, which is sensitive to outliers (i.e. non-normal data).  Alternatively, the median is less sensitive to outliers (and therefore more robust).
A great example of non-normal data when many statistics fail is bi-modally distributed data.  Because of this, it's always good practice to visualize your data as a frequency distribution (or even better, test for normality!)
A: I thought this was too obvious, until I saw this question!
When you normalise data, make sure you always have access to the raw data after normalisation. Of course, you could break this rule if you have a good reason, e.g. storage.
