An iterative square root of a self-map f is a self-map g such that $g(g(\cdot ))=f(\cdot )$ . We obtain new characterizations for detecting the non-existence of such square roots for self-maps on arbitrary sets. They are used to prove that continuous self-maps with no square roots are dense in the space of all continuous self-maps for various topological spaces. The spaces studied include those that are homeomorphic to the unit cube in ${\mathbb R}^{m}$ and to the whole of $\mathbb {R}^{m}$ for every positive integer $m.$ However, we also prove that every continuous self-map on a space homeomorphic to the unit cube in $\mathbb {R}^{m}$ with a fixed point on the boundary can be approximated by iterative squares of continuous self-maps.