So a lot's happened recently.
The dogfighting module was scrapped in December basically so they could make something better. They delivered last week with Arena Commander v0.8. To keep it in-universe it's presented as a simulator you can access through the hangar. If you have the ships it gives you an additional 'trainer' version which you can enter and activate the 'simulator'.
The currently implemented ships are the Aurora, the 300i and the F7C Hornet but if you have another ship that isn't implemented yet, (Like a Cutlass, Freelancer, or Avenger,) you get the use of an equivalent, (Cutlass/Avenger get a 300i, Freelancer gets a Hornet,)
There are only two game modes at the moment, Free Flight which is at it sounds, and Vanduul Swarm where you and two wingmen, (in F7C Hornets,) fend off waves of Vanduul, (bad guy aliens,) fighters.
They'll be adding more ships and game modes as they go.
I've been flying around in my 300i.
Haven't done any dogfighting yet 'cause my six year old PC is a little strained. I have found that Arena Commander actually runs better than the hangar module so I might give it a bit of a go.
Also at the moment they've put the two finalists for "The Next Great Starship", (a competition where teams would build ships for the game,) in everyone's hangar. They're pretty cool looking though you can't get in 'em. The one that looks like it has hair straighteners for engines won but there were a lot of really good ships made for the contest and I wouldn't be surprised to see a few of them implemented.