What are the examples of Riemannian manifolds that have zero scalar curvature but non zero ricci curvature? Is there any sort of classification of such manifolds?
To generalize Anton's comment a little, I should add that with the appropriate choice of $l$ and $k$, the product manifold $S^l \times N^k$ will have the property that you are looking for, where $N^k$ has hyperbolic $k$-dimensional half-space space as its cover. You can find the formulas for all of the geometric quantities related to these sorts of products in Chang, Han, Yang "On a class of locally conformally flat manifolds". This particular combination of manifolds can be used to construct many examples of manifolds with interesting curvature.
On a compact manifold that does not carry a metric of positive scalar curvature, every scalar flat metric is Ricci-flat. Thus on such manifolds there are no such metrics.
If a manifold carries a metric of positive scalar curvature then it also carries a metric of zero scalar curvature. I assume that for dimension at least 3 one could extend this statement to saying that it even admits a scalar flat metric with non-zero Ricci curvature, but I have no proof at hand currently. What I know for sure is: there are many manifolds having an obstruction against Ricci-flat metrics and admitting a metric of positive scalar curvature. On Ricci-flat manifolds, the first Betti number is at most the dimension, and if it is the dimension then the manifold is flat. This yields manifolds without Ricci-flat metrics and many of them carry a metric of positive scalar curvature.