Everything is over $\mathbb C$. Let $|\mathcal O_{\mathbb P^n}(d)|$ parametrize all the hypersurfaces of degree $d$ in $\mathbb P^n$ ($n\geq 2$). Let $S\subset |\mathcal O_{\mathbb P^n}(d)|$ be those singular hypersurfaces and $R\subset S$ be those reducible ones (we have this inclusion because hypersurfaces are always connected).
My question is
Is $R$ open in $S$?
Since singular quadric surfaces are always reducible, I guess for general case almost all singular hypersurfaces are reducible.
No. For example, plane cubic curves form a $\Bbb P^9$. A well-known result says that the locus of plane curves of degree $d$ with $\delta$ nodes is irreducible of codimension $\delta$ (Eisenbud-Harris give as a reference "Moduli of curves" by Harris-Morrison which I can't check unfortunately). So $S$ is $8$-dimensional. However, the space of cubics decomposing as a line and a conic is open in $R$ and has dimension $2+5 = 7$.
There is a nice discussion (and pictures!) in the book by Harris-Eisenbud 3264 and all that, starting at page 62.
In general, you can compute easily the dimension of the locus of $R$. I think that $S$ has always codimension $1$, corresponding to hypersurfaces with a single node.