Two results are proved for $\mathrm{nul} \mathbb{P}_A$, the dimension of the
kernel of the Pauli operator $\mathbb{P}_A = \bigl\{\bbf{\sigma} \cdotp
\bigl(\frac{1}{i} \bbf{\nabla} + \vec{A} \bigr) \bigr\} ^2 $ in $[L^2
(\mathbb{R}^3)]^2$: (i) for $|\vec{B}| \in L^{3/2} (\mathbb{R}^3),$ where
$\vec{B} = \mathrm{curl} \vec{A}$ is the magnetic field, $\mathrm{nul} \
\mathbb{P}_{tA} = 0$ except for a finite number of values of $t$ in any compact
subset of $(0, \infty)$; (ii) $\bigl\{\vec{B}: \mathrm{nul} \mathbb{P}_{A} = 0,
| \vec{B} | \in L^{3/2}(\mathbb{R}^3) \bigr\} $ contains an open dense subset
of $[L^{3/2}(\mathbb{R}^3)]^3$.