While I've given away the ratings already, I thought it would be useful to summarize my final thoughts. What I found over the course of writing this article is that while there is no one clear app for everyone, Linux offers many useful tools to do the job. And it's an easier job than you might expect.
The keys areas for most apps to improve is in creating a single click rip and transcode, better default settings that don't have to be changed by most users, and fixing the known bugs. If more work is put into these three areas then GNU/Linux DVD rippers will finally reach their potential.
I chose K3b as the best DVD ripper because of its clear and attractive interface and its ability to get the job done quickly with minimal fuss and configuration. dvd::rip adds the subtitle feature that is missing from K3b, but goes a step farther to be the only app I found that includes a cluster mode to harness the power of multiple machines on your home network.
If copying a DVD from one physical disk to another is your method of choice then I would look no further than K9Copy. It's simple, easy, and effective.
My final app mention is Thoggen. Its beta status clearly showed during my tests, but the idea and organization of the program makes it a strong future candidate, particularly if you are a Gnome user.