What do we know about the phase transition of #P-Complete problems?
For counting independent sets, there is a recent proof for a computational phase transition, by Allan Sly:
http://arxiv.org/abs/1005.5584
(the algorithm is by Dror Weitz from 2006; Allan proved the matching hardness and co-won the best paper award in FOCS'10 for that)

Note that for random 3SAT and similar problems there is no proof that those problems are indeed hard in the appropriate interval. When you go to the harder counting problems, it becomes easier to prove the hardness.