Why is the Twitter website unusable since ca. 2012-12-19?

Because you have AdBlock Plus enabled.

Symptoms include: no JavaScript controlled internal popup windows work any more (like the one to write a new tweet or the one to watch a profile preview of somebody). The "Follow" and "Favorite" buttons do not work either. This happens in Firefox (using Fanboy's AdBlock list version 2012-12-19) and also in Chrome. Both tested on Linux, but it should be OS independent.

To fix this, just disable AdBlock Plus for twitter.com. There are no ads anyway there. There should be a better (less brute force) fix that involves changing a rule of the filter list and sending it in as a patch, but I did not try to find this.