Is the Golden Ratio's purported aesthetic appeal supported by scientific evidence?
I found a little discussion of the issue in Russell (2000), where he summarises some of the views of the scientific literature:
Green (1995) argues that "[t]here seems to be, in fact, real psychological effects associated with the golden section, but they are relatively sensitive to careless methodological practices."
Russel (2000) explored the height to width ratio of a large database of famous paintings and did not find any particular support for a preference for the golden-section. Russel suggests that in the real world functional factors tend to constrain ratios.
McManus (1980, PDF) provides an interesting discussion of the methodological issues and the findings of research that has empirically studied aesthetic judgements of simple figures. McManus advocates the use of the paired-comparison method whereby participants make aesthetic judgements on which of series of pairs of objects is more aesthetically attractive. McManus observed that there were substantial individual differences in such preferences. McManus felt that existing empirical research was unable to adequately differentiate preference for the golden-section versus other similar ratios such as 1.5, 1.6 or 1.75.
To add a small neuroscientific point to excellent @JeromyAnglim answer - there has been an interesting study by Rizzolatti group (guy who 'discovered' mirror neurons) published in PLoS ONE. Di Dio, et al. (2007) looked at the brain responses to Classical and Renaissance sculptures, but they manipulated the proportion of sculptures' features by violating the golden ratio, as showed on the figure below (source: Di Dio et al., 2007):
During the experiment participant's were asked to either observe sculptures as if they were in a museum, or give an aesthetic/proportion judgements. fMRI scans were taken during the task. Following the results from brain activity, authors concluded that:
Therefore, the presence of the golden ratio in the sculpture determined brain activations different to those where this parameter was violated. It's quite interesting result that indirectly supports the association between positive aesthetic perception and the presence of golden ratio.