How to determine the sign of the s-wave scattering length? I guess it is relatively easy to determine the magnitude of the scattering length $a$. 
We just need to measure the scattering cross section. In this way, we can determine the value of $a^2$.
But how to determine its sign?
 A: I think the appropriate method will depend on what system you're studying.  I don't know about a general method, but since you also asked about experiments which determined the sign of $a$, I can offer a specific example:
In ultracold atomic physics experiments there is the case of Feshbach resonance, where you have a large positive scattering length due to a weakly bound state, and then as you adjust the bound state energy across the resonance (by varying the applied magnetic field) the scattering length changes sign.  As was done in this experiment, one can measure the binding energy of the molecules (at various magnetic field strengths) using rf spectroscopy, and then use this information to determine the magnitude and sign of the scattering length.
A: One way to measure the magnitude is by looking for a density-dependent energy shift. If you transfer atoms between, say, two hyperfine states with a microwave transition, the resonant frequency of this transition will change due to the mean-field shift from the interactions. If you start with an interacting mix of states 1 and 2, and you transfer from state 2 to state 3, you will find that
$\Delta f=\frac{2\hbar}{m}n_1(a_{23}-a_{13})$
This equation comes from the PhD thesis of Cindy Regal, where you may find more information in section 5.3. She used this technique to characterize a Feshbach resonance, for which it is very well-suited.
So this depends on the sign of the scattering length, but clearly it won't always be helpful: one would mainly use it if one scattering length is already known, and if they are identical it is completely useless!
I think the more general answer is rather boring: it is determined by a combination of experimental measurements of the magnitude from scattering experiments plus detailed numerical modeling. In some cases, as Mark Mitchison has said, one might tell from states of matter that depend on the sign, but I'm not aware of any cases in which this has actually been used.
I don't really know if this is the whole story or not, so I welcome any other comments.
A: This is indeed a tricky problem , which appears in several branches of physics (not just atomic physics, but also nuclear and particle physics). If all you have is two-body scattering data you need to look for interference with an amplitude of known sign. In nuclear physics, for example, you can use the sign of the Coulomb amplitude to determine the sign of the strong scattering phase shift. 
In a many-body system it is indeed easiest to look at the sign of the mean-field shift, which is directly proportional to $a$. 
