What direction does a ball move whose axis of spin is neither parallel nor orthogonal to a flat surface? Assume non-slip conditions on a flat surface. Assuming that a ball starts from a standstill, and starts spinning of its own accord, what's the relationship between the translational direction a ball moves and its axis of spin? (Think in terms of the BB-8 droid from Star Wars.) 
Google searches have shown me nothing other than simplistic situations where the axis of spin is perfectly parallel to the ground and orthogonal to the ball's direction of motion.
 A: I think the thing you are describing looks like this:

Now the point where the ball touches the surface has a velocity $v=\omega r'$. We can compute the linear distance $r'$ from $R$ and $\theta$:
$$r' = R\sin\theta$$
If the angular velocity is constant, the ball will run in a straight line, with velocity $v=R\omega\sin\theta$ in a direction perpendicular to the plane of the paper as drawn here (plane that includes both the axis of rotation and the contact point).
The first improvement to that simple model would take into account that the contact will not be a "point", but rather an "area". As drawn, points to the right of the contact "point" will be moving faster, and points to the right will be moving slower, than the central point. This results in a torque about the vertical axis, which will appear anticlockwise if viewed from the top.
When you add that vector to the vector of the rotation, it tends to cancel the vertical component - meaning that over time, the sphere will start to rotate more about the horizontal axis. Of course if there is a contact area there will be a general slowing down of the rotation (heat dissipation due to friction because not all contact points travel at the same speed).
It gets more interesting if the ball "starts to rotate" from a stationary position, because then the net force on the ball (not through the center of mass) will create a torque that will cause precession of the axis of rotation - in the diagram as drawn, the axis of rotation will tilt towards the observer, so as the ball rolls away from you it would appear to swerve to the right.
Peripherally related: this video of a concept for a new spherical wheel...
