Yes, there is a topological foliation of $\mathbb R^3$ by smooth circles. A foliation by topological circles was constructed in by Vogt in

Vogt, Elmar, A foliation of ${\mathbb{R}}^3$ and other punctured 3-manifolds by circles, Publ. Math., Inst. Hautes Étud. Sci. 69, 215-232 (1989). ZBL0688.57016.

In the subsequent paper he improved to the result, extending it to higher dimensions and getting smooth circles (but still, just a topological foliation):

Vogt, Elmar, Existence of foliations of Euclidean spaces with all leaves compact, Math. Ann. 296, No. 1, 159-178 (1993). ZBL0816.57018.

Yes, there is a topological foliation of $\mathbb R^3$ by smooth circles. A foliation by topological circles was constructed in by Vogt in

Vogt, Elmar, A foliation of ${\mathbb{R}}^3$ and other punctured 3-manifolds by circles, Publ. Math., Inst. Hautes Étud. Sci. 69, 215-232 (1989). ZBL0688.57016.

In the subsequent paper he improved to the result, extending it to higher dimensions and getting smooth circles (but still, just a topological foliation):

Vogt, Elmar, Existence of foliations of Euclidean spaces with all leaves compact, Math. Ann. 296, No. 1, 159-178 (1993). ZBL0816.57018.

But these foliations correspond to decompositions of $\mathbb R^3$ which are not upper semicontinuous. There are no upper semicontinuous decompositions of $\mathbb R^3$ in round circles, see

Cobb, John, Nice decompositions of (R^ n) entirely into nice sets are mostly impossible, Geom. Dedicata 62, No. 1, 107-114 (1996). ZBL0854.54015.