As a partial answer, there is no smooth partition of $\mathbb{R}^3$ into smooth loops.  For, consider the quotient space; it is necessarily a two-orbifold (locally a surface, possibly having isolated points each having a neighbourhood modelled on $\mathbb{R}^2$ modulo a finite-order rotation). 

Since $\mathbb{R}^3$ is non-compact and simply connected, so is the quotient.  The only such orbifold is $\mathbb{R}^2$.  Thus the partition gives $\mathbb{R}^3$ the structure of an $S^1$-bundle over $\mathbb{R}^2$.  But the fundamental group of any such bundle is $\mathbb{Z}$, contradicting the fact that $\mathbb{R}^3$ is simply connected. 

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Smoothness is not strictly necessary.  Instead we need that every loop has a small neighbourhood which is homeomorphic to a "standard fibring of a solid torus".  This obtained, the partition gives $\mathbb{R}^3$ the structure of a [Seifert fibered space][1].  The [non-compact case][2] is more difficult than the [compact case][3], but there is still a rich theory. 


  [1]: https://en.wikipedia.org/wiki/Seifert_fiber_space
  [2]: https://mathoverflow.net/questions/462959/non-compact-seifert-manifolds
  [3]: https://pi.math.cornell.edu/~hatcher/3M/3M.pdf