Lilian has a sensible commentary on the copyright case being brought against YouTube.
"YouTube only hosts material provided by third parties, and doesn't put up its own materials (as MP3.com did), it's protected by the DMCA and ECD safe harbors. (Unless a US or European court can be convinced that it had "constructive" notice of illegality - ie it should have known what was going on or as the DMCA and ECD put it, was "aware of facts or circumstances from which infringing activity is apparent" - which is not altogether impossible but perhaps unlikely.) While the Napsters of this world fell foul of secondary copyright infringement, because their central database pointed at illegal copies hosted by other users. They didn't get the benefit of the DMCA because they weren't seen as a host who could respond to NTD notices and were aware of infringing activity. This seems, in retrospect, mildly curious.
As for a Grokster analysis - as Technollama also points out, it's hard to argue that YouTube "induced" copyright infringement. Their site unlike Grokster's is free of anti-copyright rhetoric and their ToS are impeccable (not that that helped Grokster!) - plus YouTube can calmly say the site was mainly set up to allow users to host their own amateur copyright material, and , I think, prove it."