In their article Partitions of R3 into curves, the authors M. Jonsson and J. Wästlund show that space can be partitioned in unlinked unit circles or other kinds of curves.
Abstract. A general technique for obtaining partitions of $\mathbb{R}^3$ into curves satisfying various properties is presented. The method can be used to partition $\mathbb{R}^3$ into unlinked circles of radius one, which answers a question posed by Wilker [7], or into arbitrary collections of real analytic curves. We also apply the method to study the set of bijections of the open unit disk.