We study the maximum a posteriori probability model order selection algorithm for linear regression models, assuming Gaussian distributed noise and coefficient vectors. For the same data model, we also derive the minimum mean-square error coefficient vector estimate. The approaches are denoted BOSS (Bayesian order selection strategy) and BPM (Bayesian parameter estimation method), respectively. Both BOSS and BPM require a priori knowledge on the distribution of the coefficients. However, under the assumption that the coefficient variance profile is smooth, we derive "empirical Bayesian" versions of our algorithms, which require little or no information from the user. We show in numerical examples that the estimators can outperform several classical methods, including the well-known AIC and BIC for order selection.