My designs are complex enough that I want to use 4 layer boards. Top, ground plane, inner routing, and bottom routing, On the top and bottom layer, I can use a polygon fill for VCC, which helps with current carrying capacity (some) and adds more stability.
I have, and can use 2 layer boards but that is on a board that is not densely packedd. Say an L5 design with 100 pin packages and not a full load of peripherals.
No reason not to use a ground plane under the chip, but it will make routing more difficult.
Hope this helps