You cannot scale it because of the Zero-product Property. This is a mathematical theorem that Hotmath defines by saying:
The Zero Product Property simply states that if a*b=0, then either a=0 or b=0 (or both). A product of factors is zero if and only if one or more of the factors is zero.
This relates to your issue because you are trying to multiply the dimensions (let the Object Dimensions be represented by a) -- set at 0 -- by a non-zero value (let that random, non-zero value be b). As Blender follows the various laws of mathematics, the dimensions will always return to zero.
When first you add a Plane object and before you click on anything else, you should be presented with a number of settings in the Toolshelf on the left side of the 3D Viewport. Make sure the Radius: value is set to a non-zero value. If it is set to zero, it will likely spawn at zero each time you try.