Spacing out with blue 1/2" styrofoam

I don't see why not. It may provide some sound insulation. Just be sure about the spacing for doorways and trim since it will be thicker.

Why not between the studs?

Styrofoam is a compressible material and you might have tiling issues down the road. If you need to space out-Furring? Rip some 1/2 plywood and use that. You can put Styrofaom between the studs and it will reduce noise migration but if you have a full wall cavity, use R-13 fiberglass.

Inside the studs will work but you will have to work around any electrical running through the walls.

Why are you using the Styrofoam? We might be able to recommend better materials.