A corollary in The Geometry of Moduli Space of Sheaves (Huybrechts, Lehn) says:
Let $X$ be a normal closed subscheme in $P^{N}$ and $k$ an infinite field. Then there is a dense open subset $U$ of hyperplanes $H \in \left| \mathcal{O}(1) \right|$ such that $H$ intersects $X$ properly and such that $X \cap H$ is again normal.
Let $X^{'} \subset X$ be the set of singular points of $X$. Then $$\operatorname{codim}_{X }(X^{'}) \geq 2$$ If $H$ intersects $X^{'}$ properly, then $$\operatorname{codim}_{X \cap H}(X^{'} \cap H) \geq 2$$
too.
My question is: what does "intersect properly" mean? Can we find a dense open subset $U$ of hyperplanes $H\in|O(1)|$ such that $H$ intersects $X^{'}$ properly?
Definition (Stacks 0AZQ). Let $X$ be a nonsingular variety. Let $W,V\subset X$ be closed subvarieties with $\dim W=s$ and $\dim V=r$. Then $W$ and $V$ intersect properly if $\dim(W\cap V)\leq r+s-\dim X$.
For a closed subvariety $X$ in $\Bbb P^n$, the condition that a hyperplane $H$ intersects $X$ properly is exactly the condition that $H$ does not contain $X$: $\dim(X\cap H)$ needs to be at most $\dim X + n-1 -n = \dim X -1$, which happens exactly when $X\not\subset H$. For a normal closed subscheme $X$, we can say $X$ intersects $H$ properly if $H$ doesn't contain any of the irreducible (= connected) components of $X$. Combining the fact that the set of hyperplanes containing any fixed subvariety is a proper closed subset of the space of hyperplanes with the fact that we're over an infinite field, we see the union of these sets of hyperplanes is a proper closed subset of the space of hyperplanes, so the property of intersecting $X$ properly is generic.