As you can tell by the years each played, most of those players were pitches. Only Hank Biasatti, a first baseman for the Athletics, and Nelson were not pitchers for a significant portion of their careers. Charlie Cody started five games as a pitcher and also appeared in the outfield, second base, and catcher.
Another active position player who has struggled to single is Matthew Brown, a third baseman for the Angels. He batted 27 times in 2007 and 2008, hitting only a double. Brown is currently playing for the AAA Salt Lake Bees. Both Nelson and Brown will likely get more chances to single in the majors, so hopefully their time around this list is short.