I think the closest thing to what you're talking about is a Lagrange Point
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Yeah, this seems to be the closest. Or as another in the thread answered, inbetween two parent bodies. Would probably not be stable for that long, though...
Yeah, that doesn't really work. https://en.wikipedia.org/wiki/Three-body_problem
Ah yes, the hyperintelligent aliens send sentient subatomic particles to fuck everything up, don't they...
Lagrange points L4 and L5 are stable (think trojan asteroids), L1, L2 and L3 are unstable and require some propellant for station keeping (think James Webb).
You mean a tide locked planet, as in, one side of the planet would always face the star? The Moon does this with respect to the Earth, so yes, it is possible.
If you mean no spinning around the star, then no. The orbital equilibrium is given by the attraction force between the planet and the star being countered by the centrifugal force of the planet spinning around the sun. If the planet were to slow down, it would move closer to the star. If it would stop, it would eventually fall into the star.
What if something was just plopped gravitationally equidistant from two stars in a binary system?
If it was perfectly balanced, sure. Any slight imbalance would cause rapid destabilization.
Also the pure gravitational forces might just rip that thing apart.
So the stars would need to be tiny and the planet fairly massive? And how big an imbalance would cause destabilisation?
It could be stationary at the barycenter of a binary star system, but any perturbation would tend to destabilize it.
No, not really - if in geosynchronous orbit you could choose your frame of reference so it rotates with the parent body, making the satellite appear stationary, but that feels like trickery. In terms of mechanics, the satellite will be moving roughly parallel to the surface; if you took a satellite and magically stopped all its momentum, it'd drop in a straight line towards the parent. Changing the speed means changing the trajectory, probably flinging the satellite off into space or putting it on collision course with its parent.

If r was zero, or m was zero it could work.
So it’s either at the body’s core or it has no mass.
It can be above the same place on the parent body if its planet stationary but if it actually didn’t have velocity around the body, it would fall into it.
If you stuck the 'planet' at the center of mass of two (or more) orbiting stars, I guess ?
It wouldn't be a stable system, but you could use active stabilization (i.e. honking big rockets) if you had your heart set on it.
Assuming you could determine your position between the two stars with decent accuracy, you shouldn't need a "honking big rocket" just to maintain your position. A small ion engine or cold gas thruster should suffice.
Depends how big a planet is...