Let $M_{lambda}$ be the $lambda$-component Milnor link. For $lambda ge 3$, we determine completely when a finite slope surgery along $M_{lambda}$ yields a lens space including $S^3$ and $S^1times S^2$, where {it finite slope surgery} implies that a surgery coefficient of every component is not $infty$. For $lambda =3$ (i.e. the Borromean rings), there are three infinite sequences of finite slope surgeries yielding lens spaces. For $lambda ge 4$, any finite slope surgery does not yield a lens space. We also discuss generalizations of our present results. Our main tools are Alexander polynomials and Reidemeister torsions.