  This paper characterizes the possible blow-up of solutions for the 3D
magneto-hydrodynamics (MHD for short) equations. We first establish some
$\epsilon$-regularity criteria in $L^{q,\infty}$ spaces for suitable weak
solutions, and then together with an embedding theorem from $L^{p,\infty}$
space into a Morrey type space to characterize the local behaviors of solutions
near a potential singular point. More precisely, we show that if
$z_{0}=\left(t_{0}, x_{0}\right)$ is a singular point, then for any $r>0$ it
holds that $$ \limsup _{t \rightarrow t_{0}^{-}}\left(\left\|u(t,
x)-u(t)_{x_{0}, r}\right\|_{L^{3,
\infty}\left(B_{r}\left(x_{0}\right)\right)}+\left\|b(t, x)-b(t)_{x_{0},
r}\right\|_{L^{3,
\infty}\left(B_{r}\left(x_{0}\right)\right)}\right)>\delta^{*}; $$ $$
  \limsup\limits _{t \rightarrow t_{0}^{-}}\left(t_{0}-t\right)^{\frac{1}{\mu}}
r^{\frac{2}{\nu}-\frac{3}{p}}\|(u,b)(t)\|_{L^{p,
\infty}\left(B_{r}\left(x_{0}\right)\right)}>\delta^{*} \text { for }
\frac{1}{\mu}+\frac{1}{\nu}=\frac{1}{2},\,2 \leq \nu \leq \frac{2 p}{3},\,
3<p\leq\infty;$$ $$
  \limsup\limits _{t \rightarrow t_{0}^{-}}\left(t_{0}-t\right)^{\frac{1}{\mu}}
r^{\frac{2}{\nu}-\frac{3}{p}+1}\|(\nabla u,\nabla
b)(t)\|_{L^{p}\left(B_{r}\left(x_{0}\right)\right)}>\delta^{*} \text { for }
\frac{1}{\mu}+\frac{1}{\nu}=\frac{1}{2},\, \nu \in\left\{\begin{array}{ll}
  {[2, \infty],} & p\geq 3
  {[2, \frac{2p}{3-p}],} & \frac{3}{2}\leq p<3
  \end{array}\right. $$ where $\delta^{*}$ is a positive constant independent
on $\nu$ and $p$.
