For a group $G$, we are all familiar with the notion of the conjugacy class of an element $s \in G$, being the set of conjugates of $s$ in $G$, $\textit{i.e.}:$
$$\{t^{-1}st \mid t \in G \}.$$
I have seen the conjugacy class of an element $s \in G$ variously denoted by $\textit{e.g.}$ $c(s)$, $\textrm{Cl}(s)$, etc.
Is there a more or less universally acknowledged correct standard notation, as one might expect there to be for such a basic and prevalent mathematical object?
It varies with the context. In an abstract setting (such as a proof) for the conjugacy class of $s \in G$ you generally write $s^G$ or $\;^Gs$, $\operatorname{Cl}(s)$, or $C_s$, $C(s)$, and more. Watch out for confusion with the centraliser of $s$, denoted as $C_G(s)$ (you do see both objects together sometimes, as they are involved in the class equation).
Moreover, when you are interested in studying conjugacy classes in general (for example when you compute a character table) you denote them as "$nL$" where $n$ is the order of the elements in it and $L$ is a letter. So for example if you had two conjugacy classes of elements with order $2$ you would call them $2A, 2B$.
Sometimes you know more, and you employ a more explicit notation. The classic example is the one of symmetric groups: conjugacy classes are determined by the cycle structure, so you denote them with it instead: for the classes of $S_4$ you would write $(1), (2), (2)(2), (3)$ and $(4)$