Wikipedia provides examples of problems where the counting version is hard, whereas the decision version is easy. Some of these are counting perfect matchings, counting the number of solutions to $2$-SAT and number of topological sortings.
Are there any other important classes (say examples in lattices, trees, number theory and so on)? Is there a compendium of such problems?
There are many types of problems in $P$ which have $\#P$-hard counting versions.
Is there a version of a natural problem in $P$ that is more completely understood or simpler than general bipartite perfect matching (please include details on why simpler such as being provably in the lowest classes of the $NC$-hierarchy and so on) in some other area (such as number theory, lattices) or at least for particular simple bipartite graphs, whose counting version is $\#P$-hard?
Examples from lattices, polytopes, point counting, number theory will be appreciated.