Let $k$ be a perfect field, and let $$ m = \prod_{i = 1}^{n} x_i^{d_i} \in k [x_1, \dots, x_n] $$ be a monomial such that $d_i \geq 1$ for all $i$. Moreover, if $k$ is of characteristic $p > 0$, then we assume that $\gcd (d_i, p) = 1$ for all $i$.
Let $H = \mathcal{V}_+ (m) \subset \mathbb{P}_k^{n - 1}$ be the projective hypersurface generated by $m$. (Without loss of generality we can consider $x_n$ as the homogenization variable.) Denote with $H_\text{sing} \subset \mathcal{V}_+ (m)$ the closed subscheme that consists of all singular points of $\mathcal{V}_+ (m)$.
How can one compute $\dim (H_\text{sing})$ or a non-trivial upper bound $\dim (H_\text{sing}) \leq l < \dim (H)$?
Edit: For at least one $i$ one must have that $d_i = 1$, else the hypersurface is singular.
Consider $m:=x_1^2(x_2\cdots x_n)\in k[x_1,x_2,\ldots, x_n]$, then $H$ is the union of $n$ hyperplanes in $\mathbb P^{n-1}$ (1 with multiplicity 2), and $H_{sing}$ contains this hyperplane ($x_1=0$), whence $\dim H= n-1=\dim H_{sing}$.
Actually $H_{sing}$ consists of other components: the $n-2$-dimensional hyperplanes, given by the vanish of 2 coordinates ($\neq x_1$), for example $(1:0:0:x_4:\ldots:x_n)$.