Point estimation: bayesian posterior mode or posterior mean?

The posterior mode can change with reparametrization (see the common problems with MAP estimation in continuous distributions), so it's not a good idea in one of these cases (dirichlet process mixtures are a good example off of the top of my head). On the other hand the posterior mean can be completely uninformative, as you can see in chapter, section 8.5.1, of the Koller & Friedman Graphical Models book (it talks about M-projection, but M-projection with a point estimate as the approximating distribution essentially gets the posterior mean).

I'd say theoretically they're both unsound, and you're better off doing something better. In practice, choose the one that is less flawed on that specific problems (for example, with unidentifiable latent variables choose the posterior mode, if your density function diverges or there is more than one interesting parametrization choose the posterior mean).

If the posterior is symmetric and unimodal, you can use the mode (which is also less expensive than computing the mean; the latter requires integration). For multimodal or skewed distributions, it's advisable to use the mean.

Sometimes actually the posterior median is preferred over mode and mean, which tends to work well for non-symmetric distributions.

Edit: In theory, the three point estimates mean, mode, median, all are consistent (i.e., converge to the true value as the sample gets larger), asymptotically unbiased, and efficient. But in practice, the behaviors may differ quite a bit. Apart from symmetry and multi-modality, IMO, the tail behavior of the distribution is also important. For one-sided tails, the mode may not be a good choice. The mean can be problematic for heavy tailed distributions since it ends up being quite far from the locations of significant probability mass. The median is the most robust to tail behavior but can be a bit difficult to compute since you have to solve int^{theta}_{-inf} p(theta|x)dtheta = 1/2. So you may have to use numerical methods to find the median location.

The most appropriate point estimate depends on your loss function. The posterior mean corresponds to to case where you have a square loss (L2) between the true parameter value and your estimate. Posterior median corresponds to absolute loss (L1) and the posterior mode (MAP) is the L0 loss.

Obviously this will depend on your parametrization. The posterior mean of x is not the same as for log x. For this reason, the MAP estimate is some-what over-rated as the parametrization should not matter in the Bayesian context. See p. 306 of David MacKay's book for more information on this.

For purposes of prediction you should integrate out the full posterior according to Bayesian theory. However, if you must use a point estimate then the optimal one is called the Bayes' point. The point estimate that minimizes the expected (under the posterior) loss in prediction.