Let $\mathbb{R}$ be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in $\mathbb{R}^n$ given as the vanishing set of a polynomial system. This problem plays an important role for studying rigidity properties of mechanism in material designs. In this paper, we design two algorithms of whose complexities are respectively $O\ \widetilde{~}(︀D^{8n})︀$ and $O\ \widetilde{~}(︀D^{6n})︀$ for computing the real isolated points of real algebraic hypersurfaces in $\mathbb{R}^n$ of degree $D$. We also propose several heuristic optimizations to avoid the most costly computation in our algorithms in most of the cases, which makes our complexity reduced to $O\ \widetilde{~}(︀D^{3n})︀$ for those cases. These algorithms lead to an implementation which is able to solve instances which were out of reach.