Easy problems with hard counting versions 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.
One interesting example from number theory is expressing a positive integer as a sum of four squares.  This can be done relatively easily in random polynomial time (see my 1986 article with Rabin at https://dx.doi.org/10.1002%2Fcpa.3160390713), and if I remember correctly, there is now even a deterministic polynomial-time solution.  But counting the number of such representations would allow you to compute the sum-of-divisors function $\sigma(n)$, which is random polynomial-time equivalent to factoring $n$. So the counting problem is probably hard.