The amenable graph $G=(V, E)$ is a graph that satisfies the following
$$ \inf\limits_{K \subset V,\, |K|< \infty} \frac{\partial K}{|K|}=0$$
I know for example that $\mathbb{Z}^2$ is amenable and the Bethe lattice (Cayley tree) is non-amenable.
I am looking for more examples of amenable and non-amenable graphs.
Honeycomb graph, isoradial graph are amenable?
The honeycomb, the triangular lattice and any other graph with finite isoperimetric dimension is amenable. The most standard examples of the class of non-amenable graphs are rooted regular trees of degree at least $d\ge 2$: take $K$ to be the neighborhood of the root and radius $n$. You have that $|\partial K|$ and $|K|$ have order $d^n$. You could also get the $\mathbb{Z}^d$ and adding edges between any two vertexes to get a non-amenable graph.
Just from those example you can generate a variety of graphs in either class, by noticing that if you have a non-amenable subgraph, your graph is also non-amenable. And that if you "clue" different amenable graphs together, without adding "too many" edges, you will still have a amenable graph.
Another classical way of generating either class of graphs is looking at Caley graphs.
Hope it was useful.