10. Given $a, b, c, d \in[-1,+\infty)$, and $a+b+c+d=0$.
Then the maximum value of $a b+b c+c d$ is
10. $\frac{5}{4}$.

Assume $b \geqslant c$ (otherwise, replace $(a, b, c, d)$ with $(d, c, b, a)$). Then
$$
\begin{array}{l}
a b+b c+c d=b(a+c+d)+c d-b d \\
=-b^{2}+(c-b)(d+1)-(c-b) \\
\leqslant-b^{2}+b-c \leqslant-b^{2}+b+1 \\
=\frac{5}{4}-\left(b-\frac{1}{2}\right)^{2} \leqslant \frac{5}{4} .
\end{array}
$$

Equality holds if and only if $b=\frac{1}{2}, c=-1, d=-1, a=\frac{3}{2}$. Therefore, the maximum value of $a b+b c+c d$ is $\frac{5}{4}$.