Is entropy of all systems zero at absolute zero? Is the entropy of every system zero at the absolute zero?
Or is it taken to be zero at the absolute zero?
Are there systems that doesn't reach zero entropy even till absolute zero?
 A: Considering very recent debates on the subjects of temperature, entropy and the Gibbs paradox, I have been involved with, I came to realize that the third law was at best a good practical thing but that's about it.
This is at least for two reasons:


*

*Temperature is a pathological concept that leads to apparently contradictory conclusions based on the intuition of temperature we have 99% of the time. An example of that is the recent debate that has been going on because of this paper on the reality or not of negative temperatures which, according to the authors, do not make sense. The fact that negative temperatures are hotter than infinite ones is indeed puzzling and notions of warm and cold are a little bit shaken as well. However none of this arises if, instead of $T$, we speak in terms of $\beta = 1/k_B T$ where heat always flows from a small beta (albeit negative) to a bigger one. The fact that the absolute zero cannot be reached is also understandable as it is in fact an infinite $\beta$ and infinities cannot be reached. Hence, from this point of view the Third law is already weird as it states something about a notion that leads to inconsistent conclusions if we stick to it too much.

*The goal of the Third law is to define an absolute entropy scale but that cannot make sense in any practical case. I don't even want to enter the quantum realm as I believe it is not necessary to make my point. Any system with interactions will end up in a ground state at very very low temperatures (or high $\beta$ I should say) and nothing will happen after that. Depending on your definition of a state you might have some degeneracy in the ground state due to some symmetries or something along that line. Fair enough, you can simply define the relevant states as being those which are invariant under these symmetry groups and you will get zero entropy for any system of your choice. 
Now, the problem arises when you bring two different systems in contact which have nothing in common. When they were separated, you surely had to use a different definition of the notion of "microstate" to get entropy zero at low temperature for each one of them. Now when you put them together, they might interact and yield a whole new set of symmetries in the ground state. The question is then, which definition of relevant microstates do you use then so that the entropy of the whole is zero at T=0? I think that you have to redefine again a new one for this new system. 
The bottom line is that when you have a thermodynamic system and you are interested in the phase behaviour of this system, you need a reference point somewhere. If the system is always the same, this reference point can be taken as being the entropy at very low temperature and every quantity will be defined with respect to this reference as long as the system itself is not changed. As a joke/example, the idea is that the entropy of a tomato at $T=0$ has no reason to be zero when the entropy of a potato is zero at $T=0$ and vice versa.
A: Not quite. Some systems can be in their ground state and still have a nontrivial state. For example, there may be several states a system can be in, all with the same, minimum ground state energy, and the entropy will therefore be $S=k_B \log N_G$, where $N_G$ is the number of degenerate (equal energy eigenvalue) distinguishable ground states a system can be in and $k_B$ the Boltzmann constant. If these degenerate ground states are not equiprobable, the entropy is $S = N_G k_B \sum\limits_j p_j \log p_j$, where $p_j$ are the probabilities for finding the state in its $j^{th}$ ground state. Likewise an imperfect crystal will have entropy "frozen into" the deviations of the actual crystal from an unflawed version. There is a nonzero number of bits needed to specify the deviation from the unflawed version of the crystal.
The Nernst Heat Postulate, or the Third Law of Thermodynamics is sometimes rendered, "The entropy of a perfect crystal at absolute zero temperature is zero" and is clearly worded in classical terms. For a quantum system one uses the concept of the von Neumann entropy, which is essentially the $S = N_G k_B \sum\limits_j p_j \log p_j$ definition and, because it can be nonzero at absolute zero temperature, the third law is not really a useful concept for quantum systems.
A practical answer, though, is often that the third law is an excellent approximation: the quantities of entropy "frozen" into ground states are almost always utterly negligible compared with the amounts of entropy a system takes on when thermalising after the absorption of some quantity of heat pushes the system away from absolute zero.
A: Are  you asking about the caveat in the third law of thermodynamics? http://en.wikipedia.org/wiki/Third_law_of_thermodynamics
The problem with zero temperature kelvin is that it will be difficult to treat a system as entirely classically. Consider the nominal definition of entropy in terms of its microstates:
$S=\kappa log{\Omega}$
If we assume a classical description of the system, $\Omega=1$ and $S=0$, right? Well the problem is when a system is classically bound in a non-minimal energy state. Consider a particle stick in a local minimum but not a global minimum of a potential. The system might have zero temperature but this does not correspond to the minimal energy state.
In this case, we still can have several possible microstates even at zero temperature and $\Omega$ does not have to equal 1.
If we consider the quantum mechanical description of the same system, it is a little different because the particle can always tunnel out of its locally minimal energy to the globally minimal energy state. 
A: Not all real materials systems. Example - any amorphous material a-Si.  It is not at its lowest equilibrium state and is disordered...
If it could reach T=0K then it wouldn't have any kinetic energy for atoms to move so to "order" itself.
A: The short answer.
The answer if yes if ground states are always non-degenerate, and no otherwise. It is not known if ground states are always non-degenerate, therefore the answer to the question is unknown.
A little more explanation.
The question invokes temperature, and therefore it is a question about conditions of thermal equilibrium where temperature is well-defined. The answer has both practical and in-principle aspects. As a practical matter, the entropy of a system may not be zero as one cools it to very low "temperature" because it may not be possible to keep it in a strict thermal equilibrium at the lowest temperatures. But in principle the issue is, can there be more than one quantum state accessible to the system when its temperature has fallen to zero? This is equivalent to the question "is the ground state degenerate?" (because for an isolated system in thermal equilibrium all states of the same energy are accessible in principle). For most systems the ground state is not degenerate, because if it were then it would only require a tiny perturbing Hamiltonian to the lift the degeneracy. So we can only get a strictly degenerate result when some sort of constraint requires it, or rules out such small perturbations. Typically such a constraint may arise from a symmetry. But it is not known, at this time, whether such a symmetry can always be found.
