How it shows most prominently at the moment is file sharing. Setting the endless copyright debates aside, what happens is that governments and large companies want to interfere with the privacy of what citizens are doing with their own bits. They say copying is theft while citizens consider twiddling their own bits a private matter that's none of anyone else's business. The citizens don't understand that while it's de facto legal to form a sneakernet—the actual legal status probably varies from place to place but nobody has ever been sued for sneakernet filesharing because nobody else never knows about it—it's illegal to form a filesharing network over the internet.
I don't promote or demote filesharing per se: it's just the cutting edge where the future trends will show years before they land elsewhere and that's what it makes it so interesting. A marginal slice of file sharing has already moved to anonymous darknets but in a few years and after a few more bad copyright/freedom-of-speech incidents with bad publicity, there will eventually be a breakthrough and the whole filesharing activity will go underground en masse.
When the masses go for it, the capacity and availability of invisible darknets will raise in orders of magnitude. That means there will be other providers in the anonymous networks as well, websites and services. There already are some, from anonymous wikis, anonymous project pages to anonymous forums but currently those are playgrounds. That is not so in ten years: there will be a major "bazaar" going on underground. While everything is anonymous and untraceable, everything is also secure. An online bank could very well operate in the anonymous network because the traffic is already cryptographically signed, and users can enjoy strong authentication if they wish to or remain a pair of anonymous public/private keys.
At that point the traditional grasp of internet control is lost.
The institutions governing the internet and the copyright and whatnot are faced with a big dilemma: do they dare to ban and make illegal anything that's not specifically permitted on the internet and if so, how to go about it in actuality. Do they lobby for laws that only allow ISPs to let citizens connect to a http proxy that validates all traffic to be "approved"? Do they extend the charges for any use of the invisible internet that is deemed illegal, to cover all users of the invisible internet?
We're still in the shadowdancing mode but the stakes are going higher, and in at most ten years the problem of control versus anonymity will have come out in the public.
We better know what we want, at that point.