In the Schrödinger picture the state vectors evolve in time, but the operators (observables and others) are constant with respect to time.
In the Heisenberg picture the states are constant while the observables evolve in time.
In the interaction picture both the states and the observables evolve in time.
I'm fine with that. But if I encounter some problem, which of these pictures are more convenient to use for which situations? Are there any rules of thumb?
I would say that the Schrodinger picture is the natural way for humans to think, and it is perfectly applicable to most problems, probably all problems in non-relativistic quantum mechanics.
The Heisenberg picture was historically important, but actually for the problems considered (stable states of an atom) there was no evolution in time. I don't know of any other applications, but I am not expert in applications and I would not be surprised if someone disagrees.
The interaction picture is specifically useful in perturbation theory. It enables one to study the difference from the Schrodinger picture when a small interaction is introduced.
A somewhat neglected picture is described by the Foldy-Wouthuysen transformation (Foldy L.L., Wouthuysen S.A., 1950, Phys. Rev. 78, 29–36). The Foldy-Wouthuysen transformation is the standard way to derive classical correspondences from qft, but the inclusion of spin makes it more complicated than is strictly necessary. It can be simplified to the field picture which considers time evolution, but not spin. $$ |f_F(t)\rangle = e^{-iH_It}|f\rangle = e^{-iH_0t}|f(0)\rangle $$ $$A_F = e^{-iH_It}Ae^{iH_It} $$ The reason for the field picture is that field operators describe interactions by acting on the space of non-interacting states. So field operators necessarily evolve as the Schrodinger picture for non-interacting particles, which is different from the evolution for interacting particles (Haag's theorem essentially says that there are no interacting fields). To derive classical correspondences, we need to define a picture in which states evolve in the same way as field operators. I have included mathematical detail in A Construction of Full QED Using Finite Dimensional Hilbert Space
