If a subset $X\subset R^3$ is closed, locally path connected and semilocally simply connected, then for each $x\in X$, the group $\pi_1(X,x)$ is torsion-free. See my answer [here](https://math.stackexchange.com/questions/4032974/non-trivial-finite-fundamental-group/4034711#4034711). I have no idea what happens if any of these conditions is weakened.