Spin Chains - Why are eigenstates always expressed in the z-basis I was wondering why when we have spin chain Hamiltonians, like the Heisenberg model, we always express the eigenstates in the spin z- eigenbasis. 
Or maybe, I could pose my question this way - to be specific, consider isotropic Heisenberg model with ferromagnetic couplings, then when I try to 'measure' the system, what do I look for? Magnetisation? if yes, then along with direction?
 A: One measures the magnetization along the direction of magnetic field. For example if the magnetic field is along the $z-$ direction then the operators $S_{z}=$ $\frac{1}{N}\sum_{n=1}^{N}\langle \sigma_{z}\rangle$ will be the average magnetization. 
Let me edit my answer. I have given answer to the last part of the question. 
"Or maybe, I could pose my question this way - to be specific, consider isotropic Heisenberg model with ferromagnetic couplings, then when I try to 'measure' the system, what do I look for? Magnetisation? if yes, then along with direction?"
So the question was which direction. I just gave an example that magnetization along the direction of field is an important quantity. In the Heisenberg case there is a rotation symmetry and therefore average magnetizations along any direction are equally important. Typically when one is interested in symmetry breaking phase transitions, spontaneous magnetization is preferred choice, as mentioned by by Norbert for Ising model given below.
\begin{equation}
H_{I}=\sum_{i=1}^{N}\sigma^{x}_{i}\sigma^{x}_{i+1}-h\sigma^{z}_{i}.
\end{equation}
