As often as Fred drives the Mystery Machine, one might think that Fred owns it. However, that isn't necessarily the case.
Early in the 2002 live-action Scooby-Doo movie, Shaggy drives away in the Mystery Machine when Fred, Daphne and Velma decide to quit the Gang.
In the movie's next scene, Shaggy still has possession of the Mystery Machine two years after the Gang has split up.
In the 2009 made-for-TV live action movie Scooby-Doo! The Mystery Begins, the Mystery Machine originates as belonging to Daphne, although Shaggy drives it because he is the only member of the Gang who has a driver's license.