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Let me begin by quoting a well-known result of Christian Baer (see here). The result goes as follows:

Theorem (Baer): Consider a connected $n$-dimensional Riemannian manifold with Dirac bundle $S$ and generalized Dirac operator $D$. Let $h$ be a smooth endomorphism field for $S$ and $s$ be a non-zero solution of $(D + h)s = 0$. Then the nodal set $N_h$ of $s$ has Hausdorff dimension $(n - 2)$ at most.

The proof actually gives more information, I have only mentioned the salient parts that pertain to my questions, which are as follows:

(a) Are sufficient conditions known so that the $(n - 2)$-dimensional Hausdorff measure of $N_h$ is non-zero?

(b) Are there (upper and lower) estimates on the $(n - 2)$-dimensional Hausdorff measure of $N_h$ in terms of $h$?

This is mainly a reference request.

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    $\begingroup$ I can think of a rather trivial example. Assume that $n=2$. Then the underlying manifold $M$ is a surface, and if it is orientable, then $S=S^+\oplus S^-$. Assume that $h$ anticommutes with the Clifford volume element, then $D+h\colon\Gamma(S^\pm)\to\Gamma(S^\mp)$. Sometimes, the index theorem forces $\ker(D+h)$ to be nontrivial either on $S^+$ or on $S^-$. If that bundle has nontrivial first Chern class, then every harmonic section must have $N_h\ne 0$. In fact $|c_1(S^\pm)[M]|$ is a lower bound on the number of zeros, which is the $0$-dimensional Hausdorff measure. $\endgroup$
    – Sebastian Goette
    Feb 7, 2016 at 11:18

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