I asked this question on MSE some time ago but didn't get a response.

Everything can be assumed in $ \mathbb{C} $, or atleast in characteristic $ 0 $.

Consider degree $ d $ hypersurfaces in projective space $ \mathbb{P}^n $. These correspond, upto nonzero scalars, to homogeneous polynomials $ h $ of degree $ d $ in $ \mathbb{C} [ x_0, ... , x_n ] $. If $ M_d = \{ x^{\alpha} \} $, ($ \alpha $ a vector of $n+1 $ non-negative integers summing to $ d $) is the set of monomials of degree $ d $, the set $ M_d $ forms a basis for $ \mathbb{C} [ x_0, ... , x_n ]_d $, so write $ h $ as $ \sum c_{\alpha} x^{\alpha} $. Then there is a homogeneous polynomial $D(d,n) $ in the coefficients $ c_{\alpha} $, called the discriminant which satisfies the property that the hypersurface $ V(h) $ is singular iff $ D(d,n) $ vanishes. So $ D(d,n) $ itself defines a hypersurface in $ \mathbb{P}^{ \binom{d+n}{n} - 1 } $.

Question (1) : Is the hypersurface $ V (D(d,n) )$ singular always?

Here is an example, taking $ d=n=2 $, so we're looking at conics in $ \mathbb{P}^2 $. Write $$ h = ax^2 + by^2 + cz^2 + dxy + eyz + fzx $$ Then one can compute $$ D(2,2) = 8abc + 2def - 2ae^2 - 2bf^2 - 2cd^2 $$ which defines a singular cubic fourfold in $ \mathbb{P}^5 $.

The singular locus is (a slight modification of) the $ 2 $-uple Veronese embedding given by $$ [u,v,w] \rightarrow [u^2, v^2, w^2, 2uv, 2vw, 2wu] $$

We see that the singular locus corresponds precisely to the non-reduced conics, of the type $ h=(ux+vy+wz)^2 $.(As every conic, upto an invertible change of coordinates is either $ xy - z^2 $ which is isomorphic to a $ \mathbb{P}^1 $ or $ xy $ which is two intersecting lines or $ x^2 $ which is a nonreduced line.) This leads to my next question-

Question (2) : Does the singular locus of $ V(D(d,n)) $ correspond to nonreduced degree $ d $ hypersurfaces always?