What are the current best upper bounds of #P? #P is the class of counting problems for problems in NP.  In other words, a solution to #P returns the number of solutions to a particular problem in NP.

I'm wondering if there have been any studies on the worst-case behaviors of current best solutions to problems in NP.  My focus in the past has been on 3-SAT, so I am particularly interested in the time it takes to count 3-SAT solutions in the worst case.  However, I ask in general, What are the current best upper bounds for any (#P-complete) problem in #P?
One such algorithm for $\#3\operatorname{SAT}$ is due to Kutzkov.