Fix a language $\mathcal{L}:=\{=\}$, and let $T$ be the $\mathcal{L}$-theory of singleton collections. Namely the theory axiomatized by the $\mathcal{L}$-sentence $$\sigma:=\exists x((x=x)\wedge(\forall y(y=y\implies y=x))).$$
From what I can tell, there are no meaningful consequences of this theory. Aside from the size of the collection being restricted, it doesn't seem to me that I can conclude anything else universal about any potential model of $T$ that is meaningful. Hence, my question:
Are there any universal $\mathcal{L}$-sentences in $T$, aside from the obvious ones like $\forall x(x=x)$ or $\forall x\forall y(\neg y=x\implies \neg x=y)$?
(By universal $\mathcal{L}$-sentences, I'm talking about sentences of the form $\forall x\phi(x)$.)
Context: I'm trying to see if there is anything restricting a two element collection from modelling the universal theory $T_{\forall}$ of $T$, where $T_{\forall}$ is the set of $\mathcal{L}$-sentences in $T$ which are universal and their consequences.
There are tons of consequences of $\sigma$; indeed, $T$ is a complete theory, so for every sentence $\varphi$, either $\varphi$ or $\neg\varphi$ is in $T$. For instance, one very meaningful universal statement in $T$ is $\forall x\forall y(x=y)$. Or, for any formula $\varphi(x)$, $\forall x\forall y(\varphi(x)\leftrightarrow \varphi(y))$ is in $T$. The theory $T$ is in some sense "trivial" (since everything is equal!), but it has plenty of meaningful consequences.
Note also that your definition of "universal sentences" is nonstandard and uninteresting. In particular, if $\varphi$ is any sentence, $\forall x\varphi$ will be a universal sentence by your definition and is equivalent to $\varphi$ in any nonempty structure. The usual definition of universal sentence is one of the form $\forall x_1\forall x_2\dots\forall x_n\varphi$ where $\varphi$ is quantifier-free, so that the universal quantifiers at the start are the only quantifiers.
With this standard definition, models of $T_\forall$ must be either singletons or empty (assuming you allow empty models), in order to satisfy $\forall x\forall y(x=y)$. (Of course, the empty set will satisfy any universal sentence vacuously.)