Is not clear that all color-related moments of nuclei would vanish. A previous answer on the subject suggests some of them are not forbidden by symmetries.
Additionally, some nuclear experiments suggests many nuclei deviate from spherical symmetry, and is not clear to what extend this invalidates the claim that "all color moments vanish to all orders"
Are these revised models able to exclude color moments to all orders? which color moments would not be ruled out by nuclear asymmetries?
On the titular question: I'm going with "None", or perhaps "N/A".
A nucleus has a mass, $M$, a charge $q=+Z|e|$, and a spin $\vec S$.
Charge and spin together means magnetic moments:
$$ \vec{\mu} = g\vec S $$
(note: my definition of $g$ here may differ from industry standards).
An electric dipole moment (EDM):
$$ \vec{p} = g_e\vec S $$
is not allowed, by symmetry.
Under parity, the magnetic moment transforms as:
$$ {\bf P:}\vec{\mu} = g{\bf P:}\vec S $$ $$ +\vec{\mu} = +g\vec S $$
(which is the same equation as the initial one). The EDM
$$ {\bf P:}\vec{p} = g_e{\bf P:}\vec S $$ $$ -\vec{p} = +g_e\vec S $$
So parity symmetry says:
$$ g_e = -g_e$$
or:
$$ g_e=0$$
Electric dipole moments violate parity.
The reason I bring that up is because a "color dipole moment" should have the same spatial symmetries as standard EDM. A color dipole moment would represent a spatial variation of the distribution of color charge and any dependence on spin would violate parity. Nevertheless, one could consider a color-magnetic dipole moment, which would be due to color-currents or spins.
Edit: So the point of that intro was that a vector moment is not going to be related to an axial-vector property of a nucleus, which I say with the caveat that for very tiny symmetry violating things, such as the neutron EDM (https://en.wikipedia.org/wiki/Neutron_electric_dipole_moment), higher-order corrections or new physics may come into play.
Now I want to address a "color dipole moment". I don't remember ever hearing this term (and I did 2 post-docs at top tier nuclear physics labs). There are 2 barriers to it being "a thing":
in nuclear physics, we are usually describing our systems with the effective field theory that is occasionally called "quantum hydrodynamics" (QHD), a field theory that describes the interaction of matter (baryons) via exchange bosons (meson). These are the low energy colorless realizations of the underlying fundmanetal degrees of freedom: quarks and gluons. (Note: that this field is called "Medium Energy Nuclear Physics". It's not traditional nuclear physics, which looks at large many body systems such as $^{235}U$, nuclear decay, energy levels, and what not, nor is it high energy particles physics, which probes the highest energies (and shortest distance scales, and newest particles) available in the lab.)...it is somewhere in-between, and uses techniques and concepts from both.
The reason (1) is possible is because there is an empirical rule: color confinement. There is no "bare color" in the lab, and I think a "color dipole moment" (electric or magnetic) would qualify as "bare color".
The more technical statement of "color confinement" is that all observable states are "color singlets". The topic is discussed on this site at Do hadrons have color moments? I'll give similar arguments:
Since an "$SU(3)$ color singlet state" isn't the most intuitive thing, it's good to start with the more familiar spin singlet state of 2 electrons:
$$ |0, 0\rangle = \frac 1 {\sqrt 2} \big( |\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle \big) $$
We've all seen this. It's commonly described as two electrons with opposite spins. It's a superposition of eigenstates of individual magnetic quantum number. From the notation, we're inclined to say a basis state is composed of one electron with spin up and the other with opposite" spin, and mathematically it is that.
But what is up/down, i.e.: what is $z$? It's just a coordinate that we made up, and our choice of coordinates doesn't matter to nature. Had we defined the $z$ axis to point along the former $y$ axis, the state would be:
$$ |0, 0\rangle = \frac 1 {\sqrt 2} \big( |\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle \big) $$ which is identical.
If we said, "No! We're not doing the $z$ axis...we are going to work relative to the $x$ axis!". Then the state is:
$$ |0, 0\rangle = \frac 1 {\sqrt 2} \big( |\leftarrow\rightarrow\rangle - |\rightarrow\leftarrow\rangle \big) $$
which is the same thing, just with different arrows. Singlet states don't care about coordinates.
Well, the same thing is true in QCD. Since the fundamental representation of $SU(3)$ is three dimensional, we need three quarks to achieve a singlet state (or 1 quark and 1 antiquark, but let's stick to baryons):
$$ \chi_{\rm color} = \frac 1 {\sqrt 6} \Big( |rgb\rangle + |gbr\rangle + |brg\rangle + |rbg\rangle - |grb\rangle - |bgr\rangle \Big) $$
You can verify that $\chi_{\rm color}$ is invariant under any $SU(3)$ transformation and thus, is colorless.
But the point here is not that $\chi_{\rm color} $ is invariant, it's that the definition of "red" or "green" or "blue" is not physically meaningful, those are just names for coordinates in a space we made up to keep track of the math. While nature does provide this internal degree of freedom, she doesn't care about our definition of colors.
In our spin analogy, we can come in the lab and turn on a magnetic field (conveniently, in the $z$-direction), and now $z$ has physical meaning. For instance, we can take our two electron state with opposite spins, and change the phase between the $|s^{(1)}_3, s^{(2)}_3\rangle$ basis states and the new state:
$$ |1, 0\rangle = \frac 1 {\sqrt 2} \big( |\uparrow\downarrow\rangle + |\downarrow\uparrow\rangle \big) $$
has angular momentum
$$|\vec L|=\frac{\sqrt 3} 2\hbar$$
with a known projection on the $z$-axis:
$$ L_z = +1 \times \hbar $$
which has physical meaning. The $z$-axis is no longer arbitrary. I know the name of the axis is arbitrary, but the point is: there is a something there to be named "The $z$-axis", there is physically a preferred direction in space.
We can't do that for QCD. We can't turn on a "red" field. If we (read: God) could, that would leave $\chi_{\rm color}$ still in a singlet, colorless state, but red, green and blue would have unique physical meaning (well at least red would...idk if there is still an $SU(2)$ green-blue symmetry, but I digress).
So if you did have a color moment, it would have to be color-agnostic...if you do something similar in $SU(2)$, you just get the singlet state back, or an unpolarized mixed state...try it.
tl;dr You can't have a color moment.
Musings: Now if we look at barebones QHD, you might be able to have a "flavor moment". Here you have one particle, a nucleon, that has two isospin states:
$$ p = |\frac 1 2, +\frac 1 2\rangle$$ $$ n = |\frac 1 2, -\frac 1 2\rangle$$
interacting by exchanging bosons:
$$ \pi^+=|1,+1\rangle$$ $$ \pi^0=|1, 0\rangle$$ $$ \pi^-=|1,-1\rangle$$
With that, the simplest bound state, a deuteron, is:
$$ d = \frac 1 {\sqrt 2} \Big( |pn\rangle - |np\rangle \Big) $$
wait, actually that is a flavor singlet combination.
Consider a hypothetical particle, which looks just like a deuteron to the layman:
$$ d_{hypothetical} = \frac 1 {\sqrt 2} \Big |pn\rangle + |np\rangle \Big) $$
This is, again, the simplest bound state, but it is "tensor polarized"$^1$ in flavor world. We know it's hypothetical because the $NN$-interaction is iso-scalar (it doesn't care about rotations in iso-spin space) so one would expect:
$$^2{\rm He} = |pp\rangle $$ $$^2_0{\chi} = |nn\rangle $$
to be (semi)stable. There is no Helium 2 isotope nor a di-neutron. Nevertheless, there are states with non-zero isospin (flavor), e.g. the mirror nuclei:
$$ ^3{\rm H} \rightarrow ^3{\rm He} $$
where the mirror reflection occurs in iso-space, not real space.
Footnotes: [1] You edited your question to talk about higher moments, and the tensor polarization (here in isospin) made me think of the real tensor polarization of the deuteron (search term: "t20 TJNAF"). tbh, I think there is a good question to be asked regarding the spin of nuclei and the allowed higher-order moments/ physical shapes.
