What is the minimum speed detectable by redshift?

Red shifts on the order of several meters per second can be detected and accurately measured with today's technology by astronomers. That is equivalent to a velocity change of 10 miles per hour. That is in fact how some of these extra-solar planets were discovered. Once the Earth's orbital motion and the Sun's motion through space are accounted for, the velocity and direction of another object with respect to the Solar System can be determined precisely. It takes a very complex and precise spectrograph to do this, but is has been and is continuing to be done right now.