In OLS regression, I find the semipartial correlation (a.k.a. part correlation) to be a very useful indicator. When squared, it shows each predictor's unique contribution to explained variance in the outcome. If one tries several models, entering predictors in various orders, the semipartial correlation will always turn out the same by the final model. This indicator can also be a useful basis for creating Venn diagrams showing the relative importance of different predictors. Given all this, why is it cited so seldom? For example, after 6,300 questions on this site, it has never come up, nor do I see it cited in journal publications.
The notion of semipartial correlation usually arises in the context when one compares the model with a predictor and the model with that predictor removed (e.g. in the context of stepwise regression). And, because squared semipartial correlation is just a standardized form of R-square decrease, texts may find it unnecessary to mention, preferring to speak about R-square change directly instead. This is because we never compare full and reduced models abstractly, we compare concrete models, where standardization is unnecessary.