It is well-known that you can't equip the surface of the unit sphere with a singularity-free coordinate system. Physicists have called this theorem (which is important for the theory of black holes) "you can't comb the hair on a coconut."
What about one dimensional higher? That is, embed a $3$-surface of constant radius in Euclidean $4$-space (a $3$-sphere). Has it been proven that we cannot equip that surface with a singularity-free coordinate system? Is this question related to the Poincare conjecture (well, now the Poincare theorem since it has been proven)?
Yes. And you can do it two essentially different ways.
Any 4-vector can be represented as a pair of 2-vectors, whose elements can each be represented as a complex number. In that notation, it is easy to show that $$ ( e^{i\phi} \cos \theta, e^{i\psi} \sin \theta ) $$ constitutes a parameterization of the unit 3-sphere in $R^4$.
A non-vanishing tangent field to the sphere arises from this parameterization, in the form of partial derivatives with respect to the three parameters. At every point of the sphere, those three partial derivatives form the coordinate system you ask for.
If this tangent field is deemed a "right-handed" system with respect to the surface (that is, with respect to the outward normal to the surface), its negative will be a "left-handed" system. The two handednesses are analogous to the two directions to go around a circle in the plane.
As to your question about the Poincaré conjecture/theorem: this insight would form an item of background knowledge in understanding the proof of the theorem, but by itself, it doesn't get you far.