What is in #P and what is not?

For several classical nonnegative integer functions we investigate if they are members of the counting complexity class # P or not. We prove # P membership in surprising cases, and in other cases we prove non-membership, relying on standard complexity assumptions or on oracle separations. We initiate the study of the polynomial closure properties of # P on affine varieties, i.e., if all problem instances satisfy algebraic constraints. This is directly linked to classical combinatorial proofs of algebraic identities and inequalities. We investigate # TFNP and obtain oracle separations that prove the strict inclusion of # P in all standard syntactic subclasses of # TFNP minus 1.