Define a "non-singular" coordinate system on a manifold as a continuous, everywhere differentiable set of coordinates such that the determinant of the metric tensor $g_{\mu\nu}$ is everywhere positive and finite. So for example, spherical coordinates in Euclidean 3-space are singular at $\theta = 90^\circ$ and at $r=0$; but of course there are non-singular coordinate systems such as the ordinary Cartesian $(x,y,z)$. On the other hand, on the 2-sphere embedded in EUclidean 3-space, there is no non-singular system.
Answers to a recent question Can you comb the hair on a 4-dimensional coconut? have shown that there exist non-singluar coordinate systems on a 3-sphere "surface."
My question is, can somebody give me an example of such a non-singular coordinate system on the 3-sphere? The simpler the example, the better!
No compact manifold admits a global coordinate system. A global coordinate system would have to be, in particular, a homeomorphism from the manifold to an open subset of $\mathbb R^n$, which is impossible when the manifold is compact.
The mention of "singularity-free coordinate system" in the question you linked was a misinterpretation of what the hairy ball theorem says. The theorem is that $\mathbb S^n$ admits a nonvanishing vector field if and only if $n$ is odd; it says nothing about whether that vector field can be a coordinate vector field or not.
It so happens that $\mathbb S^3$ (as well as $\mathbb S^1$ and $\mathbb S^7$, but no other spheres) not only admit nonvanishing vector fields, each one admits a global frame, which is an $n$-tuple of global vector fields that form a basis for the tangent space at each point. However, once again, it is not possible to find global coordinates for which these are coordinate vector fields, by the argument I sketched above.