  The three-dimensional nature of plasmoid instabilities is studied using the
reduced magnetohydrodynamic equations. For a Harris equilibrium with guide
field, represented by $\vc{B}_o = B_{po} \tanh (x/\lambda) \hat{y} + B_{zo}
\hat{z}$, a spectrum of modes are unstable at multiple resonant surfaces in the
current sheet, rather than just the null surface of the polodial field $B_{yo}
(x) = B_{po} \tanh (x/\lambda)$, which is the only resonant surface in 2D or in
the absence of a guide field. Here $B_{po}$ is the asymptotic value of the
equilibrium poloidal field, $B_{zo}$ is the constant equilibrium guide field,
and $\lambda$ is the current sheet width. Plasmoids on each resonant surface
have a unique angle of obliquity $\theta \equiv \arctan(k_z/k_y)$. The resonant
surface location for angle $\theta$ is $x_s = - \lambda \arctanh (\tan \theta
B_{zo}/B_{po})$, and the existence of a resonant surface requires $|\theta| <
\arctan (B_{po} / B_{zo})$. The most unstable angle is oblique, i.e. $\theta
\neq 0$ and $x_s \neq 0$, in the constant-$\psi$ regime, but parallel, i.e.
$\theta = 0$ and $x_s = 0$, in the nonconstant-$\psi$ regime. For a fixed angle
of obliquity, the most unstable wavenumber lies at the intersection of the
constant-$\psi$ and nonconstant-$\psi$ regimes. The growth rate of this mode is
$\gamma_{\textrm{max}}/\Gamma_o \simeq S_L^{1/4} (1-\mu^4)^{1/2}$, in which
$\Gamma_o = V_A/L$, $V_A$ is the Alfv\'{e}n speed, $L$ is the current sheet
length, and $S_L$ is the Lundquist number. The number of plasmoids scales as $N
\sim S_L^{3/8} (1-\mu^2)^{-1/4} (1 + \mu^2)^{3/4}$.
