In this example, for my futuristic racer, how would I go about steering the ships and orienting the ships to the track, while hovering, using no euler angles at all?

You know the normal vector to the track and you know which direction you want to face along the track. If your model is such that (1,0,0) is "forward" and (0,1,0) is "up", you can build a rotation matrix whose first column is the desired forward vector and whose second column is the normal to the track. The third column can be computed as the cross product of the first two. That's probably the easiest way to do it.

However, I think I would prefer to create a solid Physics simulation of the situation and apply forces instead of setting the attitude "by hand".

That is the method I used for my first game, which used soft body physics. I composed the matrix from scratch, with its side vector just being a cross product of the forward and up vectors.

Don't you mean 1,0,0 being the side though? It should be 1,0,0 for X, 0,1,0 for Y, and 0,0,1 for Z.

Regarding a solid physics sim, I actually have a decent little solid body physics engine, but I don't use it for several reasons. Partly because I work in Flash, performance is a priority, and I have 20 to 30 ships on the track. Second, I want something more stylised. Solid body physics are more realistic than I want, and less stable.

Also the method you suggest wouldn't be good enough. It relies on just one normal right? Whatever poly is directly underneath your vehicle. I tried it, and I don't like the snapping behaviour, even with interpolation.

Say the ship is perpendicular to a right angle, the 2 raycasts on either side means you get a nice, stable orientation based on the 2 adjacent track normals. Orienting to one normal would have the pitch switching between 2 normals that are 90 degrees apart.

How would you achieve pure matrix rotations with no euler angles based on raycasts like this?