If we talk of autorouters that can achieve results competitive to manual work, high quality routers as Specctra or Expedition should be considered only for a serious discussion.
I know from many examples, that a good autorouter allows to perform a routing task in minutes up to hours that need hours up to weeks with manual routing.
There are some tasks, that apparently can't be performed satisfyingly by an autorouter, e. g, routing a low-cost PC plug-in board on a two-layer PCB.
I generally share the viewpoint, that some critical routing should be better done manually. In my designs, that's mainly power supply and analog routing. Critical digital nets can be perfectly constrained by sophisticated rules (including length, distance, from-to, cross-talk, shielding, layer specific rules). It may be sometimes easier however to route some nets manually than to design these rules.
I admit, that the same consideration may apply to supply and analog nets also, so they could be routed automaticly with suitable rules as well. But the trade-off between rule design and manual routing expense is probably clearer in this case.