It is known that for the planar restricted (i.e., one body has no mass) 3-body problem, the L4/L5 Lagrange points are stable when (and only when) m0 is at least 24.96 times as massive as m1 (or the other way around). However, these orbits exist for any mass ratios and any eccentricity. Whether they are stable is another matter. This appliet allows the user to investigate which scenarios are stable and which are not. The mathematics describing how to find the initial positions and velocities is given in Lagrange Points for Eccentric Planar 3-Body Systems.
The stability analysis for the restricted circular case can be found in almost any textbook on celestial mechanics. You can also read about it here: Linear Stability of Lagrange Points: Complex Variable Notation.
In addition to giving the usual result (with a perhaps unusual and definitely streamline development), I also consider the stability of L4/L5 in a "classical flatland". That is, in a universe where the force of gravity decays like one over distance rather than one over distance squared. In flatland, L4 and L5 are always stable. There is no condition on the ratio of the masses.