Do "shape" and "volume" in "space" have any meaning for an electron? The Standard Model (SM) predicts an "aspherical" electron due to its possesing a non-zero electron electric dipole moment (EEDM).  An experiment by Hinds (2011) placed an upper bound on the asphericity of the electron without absolutely being able to say the EEDM was zero.  Most sources I've read somewhat authoritatively say that an electron is a point and thus possesses no volume.
If the SM calls something this small "aspheric", does that logically mean that it predicts the electron possesses what we would call a shape in space and thus a volume? (Even if this shape/volume is a probabality distribution function like atomic orbitals shapes)  Does "shape" even have a meaning at these length scales due to QM?  (If quantum foam theory is right, wouldn't the electron's shape be continually buffeted and deformed by the foam thus rendering any shape/volume as "indefinite"?)  Does the fact that we have nothing smaller than an electron to probe the electron mean that we can't ever know its shape even if it has one?
I know my questions probably indicate I'm helplessly stuck in the classical world, but any insights to help are appreciated.
 A: The dipole moment, if it exists, arises from virtual particles surrounding the electron. So it's not really correct to say the electron itself has a shape. As you said, as far as we know the electron itself is a point.
I imagine most physicists believe the electron can't be a point, because we're suspicious of infinities (even infinitely small ones!) and don't believe they occur in nature. However, at the current state of development of quantum gravity it's pure speculation to suggest what actually happens at scales around the Planck length.
A: A physical electron is pointlike, which is significantly different from being a point. The reason that it isn't a point are the radiative corrections acquired in the renormalization procedure. (A ''bare electron'' is a point particle, but it has no reasonable physical properties as everything of interest turns out to be infinite. Renormalization is essential for getting meaningful answers.) 
According to QED, the physical (renormalized) electron has nontrivial form factors (generating the anomalous magnetic moment and the Lamb shift), which proves that it is not a point particle. 
The electromagnetic form factors have a classical meaning, as they give the response of the electron to a classical external electromagnetic field. 
For more details, see Section ''Are electrons pointlike/structureless?'' in Chapter B2: Photons and Electrons of my 
theoretical physics FAQ.
On the other hand, the shape of an electron is determined by its charge density, which is a function of its state, hence not an invariant property of the electron. See the Section ''The shape of photons and electrons'' in the same chapter.
