A variety of Bayesian methods have recently been introduced for performing approximate inference using linear models with sparse priors. We focus on four methods that capitalize on latent structure inherent in sparse distributions to perform: (i) standard MAP estimation, (ii) hyperparameter MAP estimation (evidence maximization), (iii) variational Bayes using a factorial posterior, and (iv) local variational approximation using convex lower bounding. All of these approaches can be used to compute Gaussian posterior approximations to the underlying full distribution; however, the exact nature of these approximations is frequently unclear and so it is a challenging task to determine which algorithm and sparse prior are appropriate. Rather than justifying prior selections and modeling assumptions based on the credibility of the full Bayesian model as is sometimes done, we base evaluations on the actual cost functions that emerge from each method. To this end we discuss a common objective function that encompasses all of the above and then briefly assess its properties with respect to three representative applications: (i) finding maximally sparse signal representations, (ii) predictive modeling (e.g., RVMs), and (iii) active learning/ experimental design. The requirements of these problems can be quite different and can lead to very restricted choices for the sparse prior and final approximation adopted. In general, we find that the best approximate model often does not correspond with the most plausible full model. Finally, we consider several extensions of the sparse linear model, including classification, covariance component estimation, and the incorporation of non-negativity constraints. While closed-form expressions for the moments needed for dealing with these problems may be intractable, we show an alternative implementation that involves transforming to a dual space using simple auxiliary functions. Preliminary results show that substantial improvement is possible over existing methods.