What does it mean to say that the electron is a near-perfect sphere? It's announced that researchers at Imperial College London has found that the electron is almost a perfect sphere. The popular articles all have a nice photo of a billiard ball, etc. It is reported that they found this by measuring the "wobble" as the electron spins (and finding none).
What does it exactly mean that the electron is a sphere? Is it the wave function that is spherical? Measuring the spin wobble brings to mind a solid object, which I think the electron surely isn't. So, what would have been wobbling if it didn't have perfect spherical symmetry?
 A: Well, one strong piece of evidence for this is the fact that the electric dipole moment
$${\vec D} \equiv \int d^{3}x' \rho({\vec r}-{\vec r_{0}})$$
of the electron is nearly zero.  The electric dipole moment would measure a reflection asymmetry in a potential charge distribution of the electron.  The fact that it takes on a zero means that the electron charge distribution is symmetric about all planes.  Now, it is still possible that the electron may have nonzero values for it's higher multipole moments, while having a zero electric dipole moment which would indicate non-sphericity as well, but as far as I know, there is no evidence for this either, and finding these moments would be an even harder experiment.  
A: There's a Nature article that describes the experiment and the results, http://www.nature.com/nature/journal/v473/n7348/full/nature10104.html, but that's behind a paywall. The experiment is described in some detail in "Prospects for the measurement of the electron electric dipole moment using YbF", http://arxiv.org/abs/1103.1566 (I've only scanned the latter, but it looks to be quite informative).
From the Nature article, "In an atom or molecule with an unpaired valence electron, the interaction of the electron EDM [Electric Dipole Moment] with an applied electric field results in an energy difference between two states that differ only in their spin orientation. This energy difference is proportional to $d_e$ and changes sign when the direction of the field is reversed. A sensitive method of measuring this energy difference is to align the spin perpendicular to the field and measure its precession rate, which is proportional to the energy difference. An alternative description of the method is in terms of an interferometer. There is quantum interference between the two spin states, and the EDM appears as an interferometer phase shift that changes sign when the electric field is reversed."
An electron is not either a sphere or not-a-sphere, but we can introduce more or fewer internal degrees of freedom into the quantum fields that are used to describe experiments results that we attribute to the electron field. Introducing different degrees of freedom has consequences for the geometrical configurations of recorded experimental results. The Nature article is explicit in saying that this is intended to distinguish between different speculative quantum field theories, "many extensions to the standard model naturally predict much larger values of $d_e$ that should be detectable". This is an experimentalists' article, however, so they link to a theory paper on the subject (which I cannot access directly). If these fields give better descriptions than the standard model of particle physics, we expect to see different, less geometrically symmetrical statistics of events.
Many of the problems of reference here can be avoided if we talk about electron fields instead of about electrons. An "electron field" is less likely to be misrepresented as spherical or not spherical, but it can be associated with (representation spaces of) space-time symmetry groups (which describe in a systematic way how something deviates from being symmetrical). Care is needed because a quantum field is a more elaborate mathematical object than a classical field, but we can loosely think of a quantum field as a way to generate probabilities that the configuration of a classical field is one thing or another at any single time, while the details of quantum measurement are such that we can't talk about such probabilities at multiple times.
A: As the comment to your post says: This spherical charge distribution picture hasn't really much to do with modern QM physics.
In fact we are talking about different quantities. The charge distribution is given mathematically by $\bar{\psi}\gamma^o\psi$ while the relevant quantities are the components of the polarization/magnetization tensor $\bar{\psi}\sigma^{\mu\nu}\psi$
The electric polarization components are zero in the current (Dirac) theory (in the local restframe) while the magnetization components are non zero. The latter represent the magnetic moment of the electron. It is the Dirac equation which mathematically describes the precession of the electron searched for in the experiment.
A non-zero electric dipole moment would give the electron the very special property of either going one of two different ways in a non-homogeneous electric field. Just like the famous Stern Gerlach experiment shows in the case of the magnetic moment.
http://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment.
Such a behavior can of course never be explained by a classical non spherical charge distribution...
A: It is probably better to talk about deviation from spherical symmetry.  If you want to think of it as a sphere then the classical radius computed from equating the mass-energy of the electron to its field
$$
mc^2 = \frac{1}{4\pi\epsilon_0}\frac{e^2}{r}
$$
and with variations on this way of computing this.  If there were some deviation from spherical symmetry it turns out to be on part $10^{16}$ of this classical electron radius.
