Szemeredi's Regularity Lemma says that every dense graph can be approximated as a union of $O(1)$ many bipartite expander graphs. More accurately, there's a partition of most vertices into $O(1)$ sets such that most pairs of sets form bipartite expanders (the number of sets in the partition and the expansion parameter depend on the approximation parameter):
There are versions of this lemma for "well-behaving" sparse graphs, see, e.g.:
What surprises me about these formulations is that they only guarantee that most pairs of sets in the partition form bipartite expanders, and these bipartite expanders may be empty. So, in general sparse graphs, it's quite possible that all edges between different parts in the partition of the vertices don't belong to an expander.
I wonder whether there are formulations that give that most edges between parts are from an expander, or whether there's no hope for such a formulation.