Roughly speaking, regular subspaces are regular Dirichlet forms that inherit the original forms with smaller domains. In this paper, regular subspaces of 1-dim symmetric $α$-stable processes are considered. The main result is that it admits proper regular subspaces if and only if $α\in [1,2]$. Moreover, for $α\in(1,2)$, the characterization of the regular subspaces is given. General 1-dim symmetric Lévy processes will also be investigated. It will be shown that whether it has proper regular subspaces is closely related to whether its sample paths have finite variation.