The magnetic field generated by a point charge is given by
$\vec B =\frac{\mu_0}{4\pi}\frac{q \vec v \times \hat r}{r^2}$
I understand that $\hat r$ denotes the unit position vector with the point charge as the origin. However I don't understand in what reference frame is the velocity vector $\vec v$ defined?
It's the inertial frame in which you wish to find the magnetic field.
update
It might help to make the definition more general. Consider an arbitrary origin in an inertial reference frame, the frame you wish to find the magnetic field. Let $\vec{r_0}$ be the position vector to the field point (referenced from the origin), and $\vec{r_1}$ the position vector of the charged particle. Let $$\vec{r}_{01} = \vec{r_0}-\vec{r_1}$$ then $$\vec{B} = \frac{\mu_0}{4\pi} \frac{q\,\vec{v}\times\hat{r_{01}}}{|\vec{r}_{01}|{}^2}$$ Now the locations of the source and field points are explicit. It also removes the charged particle from the origin. Nature doesn't care where we place the origin. Perhaps that helps.
