In quantum field theory, the Feynman/time ordered Green's function takes the form
$$D_F(p) \sim \frac{1}{p^2 – m^2 + i \epsilon}$$
and the $i \epsilon$ reflects the fact that the Green's function is not unique, but must be fixed by boundary conditions. Different choices of boundary conditions yield different placements of the $i \epsilon$, such as
$$\frac{1}{(p_0-i\epsilon + \mathbf{p})^2 – m^2}, \quad \frac{1}{(p_0+i\epsilon + \mathbf{p})^2 – m^2}, \quad \frac{1}{p^2 – m^2 – i \epsilon}$$
which correspond to retarded, advanced, and "anti-time ordered". The placement of the $i \epsilon$ determines things like the allowed direction of Wick rotation.
Quantum mechanics is simply a quantum field theory in zero spatial dimensions, so everything here should also apply to quantum mechanics. But the correspondence is a bit obscure. At the undergraduate level, Green's functions just don't come up in quantum mechanics, because we usually talk about the propagator
$$K(x_f, t_f, x_i, t_i) = \langle x_f | U(t_f, t_i) | x_i \rangle$$
instead (see here, here for propagators vs. Green's functions, which are often confused). At this level it's not as useful to talk about a Green's function, because we don't add source terms to the Schrodinger equation like we would for fields.
In a second course on quantum mechanics, we learn about scattering theory, where Green's functions are useful. But some of them are missing. The four choices in quantum field theory come from the fact that there are both positive and negative frequency solutions to relativistic wave equations like the Klein-Gordan equation, but in quantum mechanics this doesn't hold: for a positive Hamiltonian the Schrodinger equation has only positive frequency/energy solutions. So it looks like there are only two Green's functions, the retarded and advanced one, which are indeed the two commonly encountered (see here). These correspond to adding an infinitesimal damping forward and backward in time.
Are there four independent Green's functions in quantum mechanics or just two? What happened to the Feynman Green's function? Can one define it, and if so what is it good for?
Edit: see here for a very closely related question.
Best Answer
The pole prescription comes about when you solve time-dependent wave equations via Fourier transforms. In quantum mechanics that would take the time-dependent Schrödinger equation to the time-independent Schrödinger equation. However the corresponding time-dependent Green's functions do not converge when you Fourier transform them, which is how the $i\epsilon$ prescription comes about. For more detail see my answer to this question, which was also linked by the OP as related.
In quantum mechanics, a time-dependent Green function is a solution of
$$(i \frac{\partial}{\partial t} - H ) G(t) = \mathbb{I}\delta(t) .$$
The solution is not unique, therefore there are as many Green's functions as boundary conditions (BCs) you can think up. A particularly useful one is the BC
$$ G^+(t) = 0 \quad \textrm{for } t<0,$$
since it gives a solution to the initial value problem via
$$\psi(t) = iG^+(t-t_0)\psi(t_0) \quad \textrm{for } t>t_0.$$
Note that this is exactly the same in relativistic quantum mechanics/quantum field theory, only that the corresponding wave equation differs and you have to Fourier transform a few more dimensions. E.g. for the Klein-Gordon equation, which was used as an example in the question, we have
$$ (\partial^2 + m^2) \psi(t) = 0,$$
such that
$$(\partial^2 + m^2) G(t) = \mathbb{I}\delta(t).$$
Imposing the same boundary condition as above and manipulating a little bit gives the Green function in Fourier space as $$ G(p) \propto \frac{1}{p^2 - m^2 + i\epsilon}.$$
Note that this is unique now (details), the sign of $i\epsilon$ has implicitly been chosen by the boundary condition of the time-dependent Green function. I can never remember if that is called the advanced/retarded/Feynman Green's function and I think the terms also differ in the literature (e.g. in scattering theory you would call this the retarded Green function, but in QFT it seems to be the Feynman Green function). Either way it is the one that is useful for solving initial value problems.
Other people will know more about this than me. Hope the answer helps anyway.