Let $X\subset\mathbb{P}^4$ be an hypersurface of degree six given by the Pfaffian of a $6\times 6$ matrix $M$ whose entries are quadratic forms in the homogeneous coordinates of $\mathbb{P}^4$. I am interested in the singular locus of $X$.


Is it true that $Sing(X)$ is the curve defined by the $4\times 4$ sub-Pfaffians of $M$? 


In this case, is it known that $X$ has ordinary singularities along $Sing(X)$, and what is the multiplicity of $Sing(X)$ for $X$? 


What about the same questions for the following 
$$
\left(
\begin{array}{cccccc}
0 & x\_1^2 & x\_2^2 & x\_3^2 & x\_4^2 & x\_5^2\\ 
-x\_1^{2} & 0 & x\_1x\_3 & x\_2x\_4 & x\_3x\_5 & -x\_4x\_5\\
-x\_2^{2} & -x\_1x\_3 & 0 & x\_1x\_5 & -x\_2x\_5 & -x\_3x\_4\\
-x\_3^{2} & -x\_2x\_4 & -x\_1x\_5 & 0 & -x\_1x\_4 & -x\_2x\_3\\
-x\_4^{2} & -x\_3x\_5 & x\_2x\_5 & x\_1x\_4 & 0 & -x\_1x\_2\\
-x\_5^{2} & x\_4x\_5 & x\_3x\_4 & x\_2x\_3 & x\_1x\_2 & 0
\end{array}
\right)
$$
particular Pfaffian?



Singularities of Pfaffian hypersurfaces
As Sasha proved the general Pfaffian is smooth. On the other hand the special Pfaffian $X$ you wrote is an irreducible hypersurface of degree $6$ in $\mathbb{P}^4$ singular along a smooth curve $C$ of degree $20$ and genus $26$. Indeed $X$ has ordinary double points along $C$.


Your Pfaffian is indeed birational to the moduli space of $(1,11)$-polarized abelian surfaces, endowed with a symmetric theta structure and an odd theta characteristic. In this sense you can find a detailed description of this hypersurface in Lemma $2.1$ of this paper: 


M. GROSS, S. POPESCU, *The Moduli Space of (1,11)-Polarized Abelian
Surfaces is Unirational*, Compositio Mathematica 126: 1-23, 2001.


