Simple question about decoherence In simple terms, decoherence is the mechanism through which a  quantum system in superposition that interacts with the environment undergoes a quick "apparent collapse" and is no longer found in superposition. For instance, if the system is in a state $|\psi\rangle=\sum_n |\psi_n\rangle$ it will evolve into one of the basis states: $ |\psi\rangle \rightarrow |\psi_i\rangle$ . 
Question 1: is this picture accurate?  
If the picture is accurate, I do not understand it, because the new state $|\psi_i\rangle$ can also be described as a superposition of states, in a different basis $|\phi_n\rangle$ (for instance $\psi$ could be position and $\phi$ momentum). Thus, after decoherence, we can still have the system in a superposition of  states $ |\psi_i\rangle=\sum_n |\phi_n\rangle$
Question 2: Does not this remaining superposition of states contradict the very idea of decoherence as making the behavior "look" classical? 
 A: Answer to question 1 is: half-yes, half-no. Answer to question 2 is: you need to think about dynamical behaviour, not just the states.
Decoherence does half the job of solving the measurement problem. In short, it tells you that you will not in practice be able to observe that Schroodinger's cat is in a superposition, because the phase between the two parts of the superposition would not be sufficiently stable. But the concept of decoherence does not, on its own, yield an answer to the question "how come the experimental outcome turns out to be one of A or B, not both A and B carried forward together into the future?" 
The half-job that decoherence succeeds in doing is to elucidate the physical process whereby a preferred basis or pointer basis is established. As you say in the question, any given quantum state can be expressed as a superposition in some basis, but this ignores the dynamical situation that physical systems are in. In practice, when interactions with large systems are involved, states in one basis will stay still, states in another basis will evolve VERY rapidly, especially in the phase factors that appear as off-diagonal elements of density matrices. The pointer basis is the one where, if the system is in a state in that basis, then it does not have this very fast evolution.
But as I say, this observation does not in and of itself solve the measurement problem in full; it merely adds some relevant information. It is the next stage where the measurement problem really lies, and where people disagree. Some people think the pointer basis is telling us about different parts of a 'multiverse' which all should be regarded as 'real'. Other people think the pointer basis is telling us when and where it is legitimate to assert 'one thing and not both things happen'.
That's it. That's my answer to your question.
But I can't resist the lure, the sweet call of the siren, "so tell us: what is really going on in quantum measurement?" So (briefly!) here goes.
I think one cannot get a good insight into the interpretation of QM until one has got as far as the fully relativistic treatment and therefore field theory. Until you get that far you find yourself trying to interpret the 'state' of a system; but you need to get into another mindset, in which you take an interest in events, and how one event influences another. Field theory naturally invites one to a kind of 'input-output' way of thinking, where the mathematical apparatus is not trying to say everything at once, but is a way of allowing one to ask and find answers to well-posed questions. There is a distinction between maths and physical stuff. The physical things evolve from one state to another; the mathematical apparatus tells us the probabilities of the outcomes we put to it once we have specified what is the system and what is its environment. Every system has an environment and quantum physics is a language which only makes sense in the context of an environment. 
In the latter approach (which I think is on the right track) the concept of 'wavefunction of the whole universe' is as empty of meaning as the concept of 'the velocity of the whole universe'. The effort to describe the parts of such a 'universal wavefunction' is a bit like describing the components of the velocity of the whole universe. In saying this I have gone beyond your question, but I hope in a useful way.
A: Decoherence solves the pragmatical questions: You have a system, it interacts somehow a little bit with the environment, you cannot control and measure the environment, and you want to know when the interaction with the environment effectively destroys any interference effects. So, it describes how a pure state $\psi = \sum f_a \psi^a$ becomes something like $\psi =\sum f_a \psi^a \psi_{env}^a$, which, once one has to ignore the environment, is effectively a density matrix $\hat{\rho} = |f_a|^2 |\psi^a\rangle\langle\psi^a|$. Which is, in other words, the pure state \psi^a with probability $|f_a|^2$.  
What defines the basis depends on the particular physics. Roughly, if the Hamiltonian would be of type $H = p_{sys}^2 + p_{env}^2 + V(q_{sys},q_{env})$, then what is "measured" by the environment is the position of the system. So, the environment defines, via its interaction term with the system, a sort of preferred basis. 
This has, in general, not necessarily much to do with "looking classical".  Say, for an atom with may radiate away some photons - a sort of interaction with the "environment" defined by the EM field - the preferred basis will be that of energy eigenstates, thus, the first known example of something which looked everything but classical.  But, on the other hand, one can try to study the classical limit using decoherence too. 
But, even if decoherence has its value in the consideration of the classical limit, it does not solve the conceptual problem known as the "measurement problem" or "Schrödinger's cat". We obtain a collapse via decoherence only by our decision to ignore the environment and to consider only an effective state of the system. Pragmatically we have no other choice - we cannot control the whole environment. But conceptually the wave function of everything remains a wave function without collapse. And what to do with this uncollapsed wave function of the universe - to add something different, which collapses the wave function of the universe, to add something different from the wave function, like the real cat, one in dBB theory or many in many worlds - is nothing where decoherence could help. It also does not help to define what splits the world into different parts, say, into the system and the environment, or which observable is the position and which the momentum - all the information about the world provided, in the Copenhagen interpretation, by the classical part.
A: Decoherence of a single wave function  to a new state, for two particles after scattering, let us say, does not mean that the two outgoing particles are no longer described quantum mechanically. They will have new functions arising from the solution of the equations for new boundary conditions.
So 1) is correct.
It is when one describes quantum mechanically an ensemble of elementary particles that one can ask the question of approach to classical distributions. 
2) does not follow  for few body systems
I have found that the concept of the density matrix clarifies the matter of decoherence.

A density matrix is a matrix that describes a quantum system in a mixed state, a statistical ensemble of several quantum states. This should be contrasted with a single state vector that describes a quantum system in a pure state. The density matrix is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics.

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If you look at this matrix, all the quantum mechanical phases of all the many particle states make the non diagonal elements non-zero. A fully described quantum mechanical ensemble can be represented by this. 
Classical systems though, with Avogadro number order of particles entering a problem (~10^23), have no off diagonal elements in such a representation, because it is only adjacent particles that can classically affect a system , directly. Thus one reaches a classical representation when the off diagonal elements of this matrix are zero.
When does this happen? When the probability of particle i to be in a quantum mechanical phase with particle j, i.e. the ij cell of the matrix, is so small as not be measurable experimentally. In a gas for example, the quantum mechanical solutions for interactions between particles have effectively zero probability when delta(p)delta(x) become very much larger than h_bar. One can safely say that the density matrix of the gas is diagonal for classical dimensions, within experimental errors . (If one goes to the very fine  detail of the matrix, around the diagonal will be submatrices of the quantum mechanical interactions of adjacent particles .)
