For brain uploading, my money is on pure algorithms. Throw a lifetime of data on top of base human emulation software, possibly influenced by known genetic data, and then iterate through an insane number of possible brains using an insane quantity of CPU until you get a virtual brain that most closely matches up with the provided data. It'll be at best a crude approximation of the original human, but just like with MP3s, nobody will really care as long as their departed loved one feels like the same person.

The "singularity" is just a mathematical abstraction, as with dark energy or string theory. The math is highly likely to describe a real phenomenon, but anyone extrapolating further detail based on that math is basically just guessing.