A set $A$ is expressible if there exists a first order logic formula with a single free variable $\alpha(x)$ such that $V_{I,v_{x=t}}(\alpha(x))=1 \iff t \in A$, that is, $A$ contains the values I should assign to the single free variable to make $\alpha(x)$ true.
Does the expressibility of a set $A$ imply the expressibility of the singleton sets $\{a\}, a\in A$ ?
E: Let $v:VAR \to U_I$ be a valuation (with $U_I$ being the universe of discourse of the interpretation $I$).
Then $v_{x=t}$ denotes the valuation that sastisfies: $v_{x=t}(x)=t, v_{x=t}(y)=v(y), \forall y\neq x$.
If $\alpha(x)$ is a formula with a single variable $x$, then after assigning some value to $x$, the truth value of $\alpha$ under the interpretation $I$ is independent of the valuation chosen.
From the comments, you are interested in examples like the first-order theory of the additive group of real numbers. In $(\Bbb{R}, +)$, you can define the set $A = \Bbb{R}$, using the formula $\phi(x) \equiv x= x$, but the only singleton subset of $A$ that you can define is $\{0\}$ (because a definable subset has to be invariant under the automorphism that sends $x$ to $-x$).