I'm working on Morse–Kelley set theory, I guess.
Let $\mathfrak Mx$ mean that the class $x$ is a set, abbreviating $\exists C, x \in C$.
By the axiom of class comprehension, the following singleton class $\{x\}$ can be constructed:
$$ \{x\} = \{y: y = x\}. $$
However, the axiom of class comprehension requires the elements to be sets. So the following holds:
$$ \forall y, [y \in \{x\} \leftrightarrow (y = x \land \mathfrak My)]. $$
Since $$(y = x \land \mathfrak My) \to \mathfrak Mx,$$ $\neg \mathfrak Mx$ would imply $\forall y, y \notin \{x\}$, and therefore $\{x\} = \varnothing$. This is a disappointing result, in that it fails to satisfy the definition of a singleton, i.e. $|\{x\}| = 1$.
Should singletons, power sets, cardinality, etc. apply only to sets?
We like to forget that definitions are sort of equations. When you say $y=\{x\}$, you are writing an equation that states that two objects are equal to each other. Or rather, you assert that given $x$, you want to have an object $y$ satisfying the above equation.
Does $1=2$? No. Does the equation $x^2<0$ holds for any $x\in\Bbb R$? Also no.
Not every equation has a solution. Just because you can write something doesn't mean that it exists. The same is true in English, or any other language, quite obviously.
If $x$ is a proper class in KM or NBG, then $\{x\}$ simply does not exist. It makes sense as much as $\sqrt{-1}$ in $\Bbb R$, or $\sqrt2$ in $\Bbb Q$, or $\frac12$ in $\Bbb Z$, or $-1$ in $\Bbb N$. Those are all objects we can write, but do not exist in that system. So $\{x\}$ is not empty or anything, it simply does not exist.
So what can we do? We can extend KM to allow collections of classes, i.e. $2$-classes or hyperclasses, as they are known. And then we can extend to $3$-classes, and $n$-classes for all $n$, etc. But you just get a type theory at the end.