There are a variety of thought experiments that are designed to make some interpretations seem more plausible than others (e.g. the PBR experiment). The problem is that there is no argument that will ever make anybody change their mind (just like with religion or politics).
As for actual experimental outcomes, all interpretations always predict the same outcome so the differences are untestable. Otherwise we would have tested them and thrown out the interpretations that didn't fit. The exception to this is the dynamical collapse theories, which predict that the wave function collapses on its own due to some not yet known physical principle, such as quantum gravity or something. Under these theories, large scale delicate experiments (such as quantum computers, or large objects in superposition states) could expose these new effects. I see proposals pop up on arXiv occasionally, but unfortunately I don't remember any of the titles.