10. If there is a square $A B C D$ in a plane and $M$ is any point in the plane, then the minimum value of $\frac{M A+M C}{M B+M D}$ is
10. $\frac{\sqrt{2}}{2}$.

Let the side length of square $A B C D$ be $a$.
If point $M$ coincides with $A$ or $C$, then
$$
\frac{M A+M C}{M B+M D}=\frac{\sqrt{2} a}{2 a}=\frac{\sqrt{2}}{2} \text {. }
$$

If point $M$ is neither $A$ nor $C$, then $M A \cdot M C>0$.
By $(M B-M D)^{2} \geqslant 0$, we know
$$
M B^{2}+M D^{2} \geqslant 2 M B \cdot M D \text {. }
$$

In square $A B C D$, we have
$$
\begin{array}{l}
M A^{2}+M C^{2}=M B^{2}+M D^{2} \text {, } \\
\text { then } M A^{2}+M C^{2}+M B^{2}+M D^{2}+4 M A \cdot M C \\
>M B^{2}+M D^{2}+2 M B \cdot M D \\
\Rightarrow 2\left(M A^{2}+M C^{2}\right)+4 M A \cdot M C \\
>M B^{2}+M D^{2}+2 M B \cdot M D \\
\Rightarrow 2(M A+M C)^{2}>(M B+M D)^{2} \\
\Rightarrow \frac{M A+M C}{M B+M D}>\frac{\sqrt{2}}{2} \text {. } \\
\end{array}
$$

In summary, the minimum value of $\frac{M A+M C}{M B+M D}$ is $\frac{\sqrt{2}}{2}$.