In what situation would Wilcoxon's Signed-Rank Test be preferable to either t-Test or Sign Test? After some discussion (below), I now have a clearer picture of a focused question, so here is a revised question, though some of the comments might now seem unconnected with the original question.
It seems that t-tests converge quickly for symmetric distributions, that the signed-rank test assumes symmetry, and that, for a symmetric distribution, there is no difference between means/pseudomedians/medians.  If so, under what circumstances would a relatively inexperienced statistician find the signed-rank test useful, when s/he has both the t-test and sign test available?  If one of my (e.g. social science) students is trying to test whether one treatment performs better than another (by some relatively easily interpreted measure, e.g. some notion of "average" difference), I am struggling to find a place for the signed-rank test, even though it seems to generally be taught, and the sign-test ignored, at my university.
 A: Consider a distribution of pair-differences that is somewhat heavier tailed than normal, but not especially "peaky"; then often the signed rank test will tend to be more powerful than the t-test, but also more powerful than the sign test.
For example, at the logistic distribution, the asymptotic relative efficiency of the signed rank test relative to the t-test is 1.097 so the signed rank test should be more powerful than the t (at least in larger samples), but the asymptotic relative efficiency  of the sign test relative to the t-test is 0.822, so the sign test would be less powerful than the t (again, at least in larger samples).
As we move to heavier-tailed distributions (while still avoiding overly-peaky ones), the t will tend to perform relatively worse, while the sign-test should improve somewhat, and both sign and signed-rank will outperform the t in detecting small effects by substantial margins (i.e. will require much smaller sample sizes to detect an effect). There will be a large class of distributions for which the signed-rank test is the best of the three.
Here's one example -- the $t_3$ distribution. Power was simulated at n=100 for the three tests, for a 5% significance level. The power for the $t$ test is marked in black, that for the Wilcoxon signed rank in red and the sign test is marked in green. The sign test's available significance levels didn't include any especially near 5% so in that case a randomized test was used to get close to the right significance level. The x-axis is the $\delta$ parameter which represents the shift from the null case (the tests were all two-sided, so the actual power curve would be symmetric about 0).

As we see in the plot, the signed rank test has more power than the sign test, which in turn has more power than the t-test. 
