My attitude to these questions is conservative:
- the spacetime signature with one time (excluding multi-time theories);
- a positively definite Hilbert space (excluding wrong sign kinetic terms for physical fields);
- energies bounded from below (excluding a negative string coupling and similar things)
New physics is only found if you can formulate your insights in the context of the physical signature. In most cases, especially in flat space, the role of different signatures is just to continue a quantity across the complex plane of a variable. According to my definition of physics, you must continue it back to the physical signature to see "physics". It's usually obvious that you have not done anything by such a procedure. On the other hand, I still believe that the role of complexification and analytical continuation of various things will become important for our more complete understanding of black hole information issues and quantum cosmology in the future. The tricks used in the entropic principle (OVV) might be an example how the continuation could solve the vacuum selection problem.
As you can see, what I find more interesting are the questions whether the signature of various objects above - spacetime geometry; positivity of fields; positivity of the string coupling - is allowed to be flipped in the Planckian, ultrashort distance regime. This is one of the questions that has led the Bogdanoff brothers to write their somewhat controversial papers. As far as I know, no one has published "more correct" answers to some of the questions they have attempted to answer.