  It is well known that the linear stability of the Lagrangian elliptic
solutions in the classical planar three-body problem depends on a mass
parameter $\beta$ and on the eccentricity $e$ of the orbit. We consider only
the circular case ($e = 0$) but under the action of a broader family of
singular potentials: $\alpha$-homogeneous potentials, for $\alpha \in (0,2)$,
and the logarithmic one. It turns out indeed that the Lagrangian circular orbit
persists also in this more general setting. We discover a region of linear
stability expressed in terms of the homogeneity parameter $\alpha$ and the mass
parameter $\beta$, then we compute the Morse index of this orbit and of its
iterates and we find that the boundary of the stability region is the envelope
of a family of curves on which the Morse indices of the iterates jump. In order
to conduct our analysis we rely on a Maslov-type index theory devised and
developed by Y.~Long, X.~Hu and S.~Sun; a key role is played by an appropriate
index theorem and by some precise computations of suitable Maslov-type indices.
