We prove function field theorems supporting the Cohen-Lenstra heuristics for real quadratic fields, and natural strengthenings of these analogs from the affine class group to the Picard group of the associated curve. Our function field theorems also support a conjecture of Bhargava on how local conditions on the quadratic field do not affect the distribution of class groups. Our results lead us to make further conjectures refining the Cohen-Lenstra heuristics, including on the distribution of

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... rtain elements in class groups. We prove instances of these conjectures in the number field case. Our function field theorems use a homological stability result of Ellenberg, Venkatesh, and Westerland.