I've got some some home movies that look great, but my video camera doesn't have one of those fancy new stabilizers. Is there extant end-user software (preferably FLOSS) I could use to removing the shaking?
You can deshake a video with the use of an external stabilizer plugin available here:
After installation of the plugin you can stabilize a video with the following command:
transcode -J stabilize -i inmovie -y null,null -o dummy
This will give you an output file with definitions for transforms that will have to be performed as a second step (here using default options):
transcode -J transform -i inmovie -y xvid,null -o outmovie
For further details and many addtitional options also see: