This presentation outlines Bayesian selection methodology for semiparametric components in two different scenarios. The first is in models that involve additive semiparametric function estimation, while the second is in time-space varying coefficient models. While these statistical models may differ, the approach to modeling the flexible components involved is similar. It entails the adoption of proper shrinkage priors, coupled with a point mass at zero. In effect, this corresponds to adopting both traditional Bayesian regularization and model averaging simultaneously. Some consideration is given to the development of efficient sampling schemes through the adoption of non-degenerate priors on the shrinkage/smoothing parameters for each component. In both cases, the methodology is targeted to enable the estimation of high-dimensional problems. The resulting applicability of the methodology is highlighted with two substantive examples. The first is a high-dimensional longitudinal model for the intra-day demand for electricity. In this example, a seemingly unrelated regression model with 288 unknown component functions is estimated. The second example is a hedonic model for the price of residential housing in a region in Sydney. The data are observed at a daily level over one calendar year and over a high resolution lattice. Here, a varying coefficient model with 30 unknown space and time components is estimated.