According to the Wikipedia page on the electron:
The electron has no known substructure. Hence, it is defined or assumed to be a point particle with a point charge and no spatial extent.
Does point particle mean the particle should not have a shape, surface area or volume?
But when I searched Google for the "electron shape" I got many results (like this and this) that says electrons are round in shape.
As far as we know the electron is a point particle - this is addressed in the question Qmechanic suggested: What is the mass density distribution of an electron?
However an electron is surrounded by a cloud of virtual particles, and the experiments in the links you provided have been studying the distribution of those virtual particles. In particular they have been attempting to measure the electron electric dipole moment, which is determined by the distribution of the virtual particles. In this context the word shape means the shape of the virtual particle cloud not the shape of the electron itself.
The Standard Model predicts that the cloud of virtual particles is spherically symmetric to well below current experimental error. However supersymmetry predicts there are deviations from spherical symmetry that could be measurable. The recent experimentals have found the electric dipole moment to be zero, i.e. the virtual particle cloud spherically symmetric, to an accuracy that is challenging the supersymmetric calculations.
However there are many different theories based upon supersymmetry, so the result doesn't prove supersymmetry doesn't exist - it just constrains it.
The shape of a distribution of charges is described in terms of multipole expansion, which you can think of as similar to Fourier expansion but in two dimensions. The total charge gives you the "monopole term", whose interaction is spherically symmetric. If there's an offset between the center of the mass distribution and the center of the charge distribution, you have a dipole moment. A coin-shaped or cigar-shaped distribution has nonzero quadrupole moment, a pear-shaped distribution has an octupole moment, and so on. As in Fourier analysis, it's possible to represent any charge distribution in terms of multipole moments, though a shape with sharp edges (like, say, a cube) would require an infinite number of terms.
The electron cannot be cube-shaped, or even coin- or cigar-shaped, due to a theorem relating multipolarity and spin. A spinless particle may have a monopole moment, but not a dipole moment; a spin-half particle may have monopole and dipole moments, but not a quadrupole moment; a spin-one particle may have monopole, dipole, and quadrupole moments, but not octupole moments. A handwavy, cartoony way to think of this is that any such moments must be quantized along the direction of the particle's spin — otherwise, as the particle spins, they'd average to zero. If you wanted the electron's charge distribution to be cigar-shaped, like a uranium nucleus is, you'd need to specify that a polarized electron has more charge near its poles than it does near its middle. But a spin-half particle doesn't have any spin projection near its middle — there are only "up" and "down." An electron may have monopole and dipole moments, but doesn't have enough degrees of freedom to have any more complicated shape.
Furthermore, we have the observation that the electron's interactions are very nearly invariant under the symmetries of parity conjugation, $P$, and charge conjugation, $C$. This further restricts the moments that are available, because the electron's spin, to which the dipole moments must be coupled, is an axial vector and does not change sign under $P$. To a very good approximation, then, the electron's mass and charge distributions may carry only a monopole moment, while its magnetic field (another axial vector quantity) may carry only a dipole moment. This gives us the usual toy-model picture of an electron as a spherical, spinning bar magnet.
However, the electron's interactions are not quite invariant under conjugation of parity and charge at the same time. This transformation, $CP$, is the operator that transforms an electron into a positron. Our strongest evidence that the universe treats electrons and positrons differently is that the universe is quite full of electrons, but contains only incidental positrons. To reach this state requires, among other things, $CP$ violation. But essentially every model that contains enough $CP$ violation to predict our observed matter/antimatter asymmetry also predicts permanent electric dipole moments for the proton, electron, and neutron which are much larger than the current limits. This is what the Hudson and DeMille groups have measured in the news stories you found. I thought that DeMille's explanation in your first link was quite nice.
Electrons, and such small things :-) are handled by quantum mechanics. Quantum mechanics differs very, very much from the classical, Newtonian mechanics and from our intuition based on our experience.
In QM, although the electron is handled as if it were a point-like body, it doesn't have an exact location. Instead of it, its location is described by a wave function named $\psi(r)$. This is a complex scalar field interpreted in the space, thus we can describe this as $\mathbb{R}^3\rightarrow\mathbb{C}$. What makes the picture really interesting, this wave function has complex values. The quadratic absolute value ($\psi\psi^*$) is the same is the probability distribution of the location of the electron on a specific place.
The integration of $\psi\psi^*$ on a volume gives the probability to the electron exists in that volume.
As a classical intuition we could imagine that as if the electron were some like a "cloud", with different densities in the space. As a possible interpretation of the "shape of the electron", we can imagine the wave function or the probability distribution, or we could even imagine this "cloud".
Well, we could even calculate this, although they aren't the simplest calculations. And from the calculated density images, we can generate visible pictures. So:
These are electron shapes around atomic nuclei. But there are also very different distributions as well, for example a free-electron in a double slit experiment has a very different wave function.
No one has ever directly seen an electron, and it's quite possible that no one ever will. To think of it as a shiny little pinball is as much a mistake as to think of it as an abstract infinitesimally small "point" with certain properties. To add to the confusion, depending on how you are "looking" at the electron, it can appear to be a particle (implying some finite size and a definite shape) or it can appear to be a wave of some sort. As a wave, you can talk about the "clouds" of electron orbitals around an atom, which are not physical things but representations of probabilities. Looking at an electron as a pinball or as a wave/cloud can be useful in certain situations, but is not an absolute truth.
Short answer: no, electrons do not have a "shape", at least in the sense of "it looks like a pinball or ...".
What is your definition of shape? And on what scale does this information bare relevance to you? If I were to tell you electrons are actually shaped like pyramids, in what way would this change the way you interact with the universe. Albeit analogies being the cause for questions like these (subatomic particles often being depicted as distinctly colored spheres), I will try to point out exactly that by using an analogy.
Buildings can be classified by type. Like a skyscraper, church, castle, bungalow whatever. What would be the smallest structure that assigning the property building type would still make sense to you. What kind of building would you call a single brick? Sure I could imagine a biologist studying ant colonies assigning this property to a brick, because in that context it is relevant information.
All those depictions addressed in previous answers only bare relevance to their respective fields of study and have nothing to do with your concept of shape. So instead of arguing about the definitions of shape you should ask yourself is this a relevant question for you?
On the other hand though, suppose you were to be competing in a regular bowling match and you were to be told to only use cube-shaped balls instead of spherical ones. Even without having tried a cube-shaped bowling ball, it's safe to say this would have (at least some) significant influcnce on how you interact with the ball.
An electron is not a point particle. Point particles don't exist. The world is governed by quantum mechanics, which describes physical systems in terms of quantum mechanical observables, which are represented by Hermitian operators. Different observables represent different ways in which you can interact with a given system and copy information from it. For example, a photomultiplier tube may be useful for telling whether there are more than N photons' worth of energy in some region for some value of N.
If you consider some finite region you can measure an observable that will give you information about whether there is an electron in that region. But that region can't be arbitrarily small. One limitation is that measuring in a smaller region requires putting more energy into that region and at some point the energy required to do that is so large it creates a black hole. There may be other physical limitations that would kick in before you get to that level.
How should we interpret claims about the electron's shape? Such a claim means that when we measure whether an electron where an electron is with some high accuracy, we get a spherical distribution of results to some accuracy sufficient to rule out some supersymmetric theory.
You might think something like "Couldn't we say that the electron really is at some particular point but not at others and we just can't tell exactly where it is?" That idea doesn't match reality because if you want to predict the electron's subsequent evolution you have to to account of observables that don't represent the electron as being at one particular point, like momentum, since those observables appear in the Hamiltonian.
The claim that the electron has no known substructure is correct but what it means is not that the electron is at a particular point but just that it doesn't have subsystems that can be changed independently. A composite system, like a biro, doesn't have that property. You can take the ink tube out of a biro and move it around independently of the plastic shell. But you can't do anything analogous with an electron as far as anybody knows.
For some relevant material see
http://arxiv.org/abs/1204.4616
http://arxiv.org/abs/quant-ph/9906007
Yes, electrons, as quasi-particles created by the environement, have state-dependent shapes (and other state-dependent properties).
In QM a shape can be well defined in scattering experiments. For example, the elastic scattering of a fast charged particle from an atom involves non-perturbed atomic wave functions, and for Hydrogen as a target, for example, one finds the famous $a_0$ with some kinematic variables in the cross section.
Similarly for the electron as a target, one obtains some cross section, but this time inelastic or even more precisely, an inclusive one since those virtual particles that "dress" the real electron are so weakly bound to it that the purely elastic cross section is zero - it is impossible to push an electron and not to excite its "dressing". As the dressing depends strongly on the environmental sizes (properties), one may obtain different inclusive pictures. For example, a "Coulomb center" is a particular inclusive picture. For partially inclusive cross section there are some factors depending on the detector resolution (also an environement feature).
Many peoples do not understand the meaning of the wave function and especially its boundary conditions as simplified solutions of a huge QM environmental problem and concetrate their attention on one excitation thinking of it as of something separate and unchengeable ("fundamental").
