If we assume that light survives the trip all the way to the singularity without interacting with anything else or some crazy thing happening, then yes, it will be severely blue shifted. I'd guess that green light would be blue shifted so far it would be in the extreme gamma ray energy range.
However, I will point out that our knowledge of what's behind the event horizon of a black hole, and especially of the singularity at the middle is extremely speculative. We don't eve know if there is a "real" singularity or if it's simply a result of an incomplete theory that doesn't work at that level of gravity. So take what I said above with a grain of salt and the knowledge that we simply don't know.