I'm interested in graph families which are sparse, and by sparse I mean the number of edges is linear in the number of vertices. |E| = O(|V|). Besides non-trivial minor-closed families of graphs (these turn out to be sparse), I don't know any other families. Can anyone suggest any interesting graph families (which are not minor-closed) which are sparse?

Please don't suggest the trivial family ("the family of sparse graphs").

EDIT: Thanks to the first few people who replied, I realized that bounded degree graphs (max degree < k) also form an interesting and large class of sparse graphs. So perhaps I'll refine my question to exclude those too. Any interesting sparse graph families where the max degree isn't bounded? For example the family of star graphs is sparse and not bounded degree. (But they're minor-closed.)