If an orbit is only stable for a very narrow range of parameters, then it’s not stable. Eventually, something would perturb it out of that narrow range, and it would then be unstable and do something crazy.
I didn’t realize bodies at the Trojan points describe corkscrew-like orbits. NASA just found a Trojan “satellite” of the Earth.
Here is a video depiction of its orbit. This seems close to qualifying as a “cool” orbit. 
Trojan orbits aren’t just an object sitting at the lagrange point. They generally oscillate around a lagrange point. This one oscillates above and below the ecliptic as well as towards and away from the Earth.
A good analogy for stable vs. unstable would be marbles on an uneven surface. If it’s a dip (some Lagrange points) the marble will roll around in the depression, provided it does not get seriously disturbed by something and fly off. An unstable point is like a marble on a hill. You can position it at the top, it may even sit there for a few seconds, but the least disturbance it will go flying off and will neve come to rest at that point.
A binary system can have stable planets if the stars are far enough apart to not wreck the stability of close planets (I.e. the sun does not disturb the earth-moon or jupiter systems); or the two stars are so close that planets orbit the combined system far enough away to be in a stable orbit. Depending on the the size/brightness of the stars, it may be possible for these planets to be in a star’s/stars’ habitable zone, the distance where sunlight is just right amount to support life.
There was some news article a few weeks ago about a new orbital observatory that would be looking for brown dwarfs and failed stars (i.e. bodies at least 10 times larger than Jupiter) and they thought based on the size and distribution of stellar masses, there was a good chance of finding one or more less than 4ly away.
Part of the coolness comes from viewing it in a rotating reference frame where the sun and earth are both stationary, though. Viewed in a near-inertial sun-stationary reference frame, these orbits look a lot more conventional.