Is there a simple example of a transitive vector field on the three sphere? I was just thinking about the transitive field on the torus and the possibility of defining a vector field without singularities on $S^3$ and this idea popped up.
Edit: I think my question was misunderstood. I am specifically asking about transitivity, or the existence of a dense orbit (flow). I know about the existence of a vector field without singularities (which is what every answer is mentioning), I just worded my initial post poorly. What I want to know is if it is possible to define a vector field on $S^3$ that has a dense orbit.

You can define a nowhere-zero vector field on any odd-dimension sphere. For instance, one is given by embedding the $(2n-1)$-sphere as the unit sphere in $\Bbb R^{2n}$, and at the point $x = (x_1, x_2, x_3,\ldots,x_{2n})\in S^{2n-1}$ define the tangent vector
$$
v_x = (x_2, -x_1, x_4, -x_3, \ldots, x_{2n}, -x_{2n-1})
$$
This is readily seen to be orthogonal to $x$, and therefore tangent to the unit sphere at $x$. At the same time, it is never zero, because the origin of $\Bbb R^{2n}$ is not part of our sphere.
On an even-dimension sphere, it can never be done, for the same reasons that it cannot be done on $S^2$ in particular. For instance, my proof here (stolen from theorem 2.28 in Hatcher, where my above example for odd-dimensional spheres can also be found) covers all of those cases simultaneously.
