The resource I'm learning forking and dividing from (Tent and Ziegler) feels slightly light on examples to me. For instance, skimming through the chapter that introduces the concepts reveals only two concrete examples of the notion.
Example: For any $b_1<b_2\in\mathbb{Q}$, $b_1<v<b_2$ divides over $\emptyset$. On the other hand, the type $\{x>a:a\in\mathbb{Q}\}$ does not divide over $\emptyset$.
Exercise: Define the cyclical order on $\mathbb{Q}$ by $$\operatorname{cyc}(a,b,c)\iff(a<b<c)\vee(b<c<a)\vee(c<a<b).$$ Show that $(\mathbb{Q},\operatorname{cyc})$ has quantifier elimination, that $\operatorname{cyc}(a,x,b)$ divides over $\emptyset$ for any $a\neq b$, and that the (unique) type over $\emptyset$ forks over $\emptyset$.
Given that (as I understand it) one of the main utilities of these notions is in classifying classes of theories, it makes sense to me that there's an emphasis on general results rather than specific examples. However, I do find myself wishing for slightly more (exercises on) concrete examples; I want to be able to look at a formula or a type and immediately have some intuition as to whether it forks or divides over a particular parameter set, and (personally) I find that the best way of building such intuition is by working through many examples. I quite enjoyed the exercise above, for instance.
To this end, does anyone know of a worksheet or other resource that contains a large number of concrete examples of these ideas? I'm thinking of something vaguely along the lines of the book "Counterexamples in Topology", containing (for instance) a number of formulas and types in specific structures, and perhaps exercises on whether they fork or divide over a particular parameter set. If my skim-through of Tent and Ziegler was also uncharitable, and their later sections do contain more examples, I'd also be happy to hear this too. Hopefully this isn't too ill-posed or vague of a question; please feel free to close if it is!
You could 'create' these exercises by yourself by looking at https://forkinganddividing.com/. It lists a whole lot of theories and places them in the classification picture. For a lot of theories there is a description of what forking (or dividing, but they coincide in the well-behaved cases, e.g. in simple theories) is in that theory. Just click on the theory and then click on characterization of forking in the bottom right box.
For example, in the theory of infinite sets we have that $\operatorname{tp}(A/BC)$ does not fork over $C$ precisely when $A \cap B \subseteq C$. You could try to prove this. In general: the more stable the examples are, the easier it will be to work things out. Strongly minimal theories are definitely the easiest, and the nonforking relation there is given by the pregeometry, something that Tent and Ziegler also covers.
Just a heads-up (although Tent and Ziegler also cover this): it is really the negation of dividing / forking that is the 'good' notion. Nonforking gives us a notion of independence, and we write this with an anchor symbol. See for example definition 7.2.8 in Tent and Ziegler. This notation is standard and the website I mentioned also uses this.