Do any fundamental particles with gravitational or electric dipoles exist? For magnetism:
$$ \oint \vec{B} \cdot \mathrm{d} A = 0 $$
For electricity:
$$ \oint \vec{E} \cdot \mathrm{d} A = \frac{Q}{\epsilon_0} $$
For gravity:
$$ \oint \vec{g} \cdot \mathrm{d} A = -4\pi G m $$
We know magnetic dipoles (single electrons) exist and yet there is no magnetic charge.
Now electric dipoles composed of multiple particles exist but I don't know of any fundamental electrical dipoles.
Do any fundamental particles with gravitational or electric dipoles exist?
I can't see anything that forbids them.
I guess more complicated cases such as photons also exist also which have oscillating electric and magnetic fields. I'm not sure how to properly describe more complicated cases like these. I'm not sure if it is right to call a photon a monopole or a dipole but I guess it's important to note that things can be more than just monopoles or dipoles.
 A: Fundamental particles like the electron are well-known to have magnetic dipole moments related to their spins.  Moreover, the standard model also predicts that they should have small electric dipole moments (EDMs) as well.  However, the EDMs predicted in the standard model are very small; they will probably be too small to observe directly for quite some time.  The reason is that having a permanent EDM violate time-reversal symmetry (T).  The standard model does have T violation, but it can only appear through quantum processes that involve at least three different types of virtual particles, and these higher-order quantum corrections are generally strongly suppressed.
There is quite a bit of interest in the idea that there may be new physics beyond the standard model that involves much stronger sources of T violation.  These would, correspondingly, produce much larger EDMs through quantum corrections.  Searches for electron and neutron EDMs are some of the best way to test these theories of new physics.
With gravity, the situation is different.  It is not actually possible to have a gravitational dipole.  An electric dipole involves a separation of the positive and negative charges in an object.  However, there are no negative gravitational sources; all masses are positive.  The lowest-order multipole for a gravitational field is the quadrupole.  For a fundamental particle to have a quadrupole moment, it must have a spin of at least 1.  (In contrast, to have a dipole moment, a particle must have a spin of at least 1/2.)  A spin-1 particle like the deuteron (a bound state of a proton and neutron) does indeed have an electric quadrupole moment.  The deuteron should also have a (T-violating) magnetic quadrupole moment and a gravitational quadrupole moment, although these would both be extremely tiny.
