The Kazdan-Warner trichotomy states that for $n\ge 3$, a compact $n$-manifold falls into one of three categories:

(A) Every (smooth) function is a scalar curvature.

(B) The manifold is strongly scalar flat.

(C) The manifold only admits negative scalar curvature metrics.

Of course class (A) is nonempty in all dimensions because it contains $S^n$. Gromov and Lawson showed that (B) contains all tori $T^n$. However, it's not clear to me that (C) is nonempty in all dimensions. Kazdan and Warner (Prescribing Curvatures, Proc. Symp. Pure Math. 27) showed:

Let $M$ be a spin manifold with $\hat A(M)\ne 0$ and $b_1(M)=\dim M$. Then $M$ does not admit a metric of zero scalar curvature.

Consequently, any such manifold must be type (C). They only give the example $T^4\#K3$. Are there examples in dimensions $3$ and $\ge 5$ of type (C) manifolds? Presumably one could use the Kazdan-Warner result above and then apply some knowledge of manifolds with nonzero A-roof genus. They mention Hitchin told them one can strengthen the hypothesis to $b_1(M)\ne 0$.