Quantum mechanics is a funny thing. On the one hand it makes fantastically accurate predictions, while at the same time it tells us that some things are either unknowable, or only knowable with a certain probability. Thus, if you ask a physicist what an electron bound to a nucleus looks like, you will probably get a number of answers that depend on who you ask. We can measure the energy, spin and orbital angular momentum of these electrons but we can not measure their position, and hence we have no idea of the sort of trajectories the electron takes as it moves around the atom. Therefore, the honest answer is probably best given by "We don't really know." However, now that we have the capability of generating light pulses that short compared to how fast we think electrons move this might be about to change.
This line of research is kind of long in fruition so allow me to give you a bit of the history behind these developments. A particular type of laser generates trains of pulses that are very short, only a few femtoseconds (10-15) in duration, however, this is still much longer than it is expected for an electron to orbit the atom. The breakthrough came when we developed the optics to shape the pulses so that the bumps and dips were much shorter than the expected orbital time for an electron. If these pulses are used to irradiate a group of chemicals that can react in different ways, then, under the right circumstances, the pulse shape can dictate which reaction occurs. This can only be explained by thinking about how an electron moves around within the constraints of the chemical bond. From these experiments we know that the electron is, under some circumstances, quite localized. However, with the development of techniques that produce ultraviolet light pulses that are about 100 attoseconds (10-18) long it may now be possible to "look" at electrons within their orbit.
How does the experiment work? Well essentially you need two electrons that are in correlated states and two laser pulses. The first laser pulse removes one of the electrons from the molecule and puts it into a known state. The second pulse, depending on the timing will either cause the molecule and electron to recombine or accelerate the electron further away from the molecule, where it can be detected as a current. Since the electrons are correlated if we know the state of one we know the sate of both. Therefore if we take a bunch of molecules and measure the number of electrons produced as the delay between the two pulses is varied, we should build up a picture of where the electron inside the molecule is and how fast it is moving.
Now at this point I should note that the experiment has not been successfully performed yet. These short pulses in the correct wavelength range have only become available in the last few years, and the people who generate them (laser physicists) don't know much about chemistry. However, I hope to be writing about this soon – with pretty (false color) images of electron wave packets orbiting a big blob.