The tautological ring of M̄g,n is rarely Gorenstein

We prove that the tautological rings R^*(M̄g,n) and RH^*(M̄g,n) are not Gorenstein when g≥ 2 and 2g+n≥ 24, extending results of Petersen and Tommasi in genus $2$. The proof uses the intersection of tautological classes with non-tautological bielliptic cycles. We conjecture the converse: the tautological rings should be Gorenstein when $g=0,1$ or g≥ 2 and $2g+n<24$. The conjecture is known for $g=0,1$ by work of Keel and Petersen, and we prove several new cases of this conjecture for RH^*(M̄g,n) when g≥ 2.