An element of Lie algebra $L$ is called $\textbf{regular}$ if dimension of its centralizer is equal to rank of Lie algebra $L$. I work over field of characteristics zero. In case of $G_2$, regular elements are those that have $\dim(C(x)) = 2$, where $C(x)$ denotes centralizer of $x$. I know some examples of regular elements in $G_2$, for example $x_{\alpha}+x_{\beta}$, $x_{-\alpha}+x_{-\beta}$, some linear combination of $h_{\alpha},h_{\beta}$ is bound to be regular. Here, I use notation of Chevalley Basis of $L$, where $L$ is a simple Lie algebra of $G_2$ type : $\{x_{\alpha},x_{\beta},x_{\alpha+\beta},x_{2\alpha+\beta},x_{3\alpha+\beta},x_{3\alpha+2\beta}, x_{-\alpha},x_{-\beta},x_{-\alpha-\beta},x_{-2\alpha-\beta},x_{-3\alpha-\beta},x_{-3\alpha -2\beta},h_{\alpha},h_{\beta}\}$ is the Chevalley Basis of $G_2$. Now, I am not aware of any paper or book that considers "classifying" regular elements of $G_2$, nor I would know where to look.
What I tried (the process is arduous) is using programming language GAP to determine intersections of centralizers of elements of Chevalley Basis, as every linear combination of two elements has to commute with everything in the intersection of centralizers of those two elements, hence dimension of centralizer of any linear combination of two elements is bigger or equal then dimension of intersection of centralizers of those two elements. Hence, if I get $\dim(C(x) \cap C(y)) \geq 3$ for some $x,y$ in the Chevalley Basis, then no linear combination of two is regular.
The flaw of this method is painstakingly obvious: you get pairs with intersection of centralizers equal to 1 or 2, but that does not guarantee you whether there actually is a linear combination of those two elements $x,y$ that has dimension of centralizer equal to 2, hence for all those pairs you have to check yourself for which $a,b$ is $\dim(C(ax+by)) = 2$,if there exist such $a,b$, where to my knowledge there is no method of saying at least when such $a,b$ do not exist.
Also, this is just the start of the issue: what happens with the combination of three or more elements of Chevalley Basis? Since I have no idea and no knowledge of this particular problem, every constructive help, book, paper, is very welcome. Thankies!