We investigate certain analytical properties of the free $α-$stable densities on the line. We prove that they are all classically infinitely divisible when $α\le 1$, and that they belong to the extended Thorin class when $α\leq 3/4.$ The Lévy measure is explicitly computed for $α=1,$ showing that the free 1-stable random variables are not Thorin except in the drifted Cauchy case. In the symmetric case we show that the free stable densities are not infinitely divisible when $α> 1.$ In the one-sided case we prove, refining unimodality, that the densities are whale-shaped that is their successive derivatives vanish exactly once. Finally, we derive a collection of results connected to the fine structure of the one-sided free stable densities, including a detailed analysis of the Kanter random variable, complete asymptotic expansions at zero, a new identity for the Beta-Gamma algebra, and several intrinsic properties of whale-shaped densities.