The Premier League (and their legal strong-arming outfit Football DataCo) have plenty of previous for shakedowns with a flimsy legal basis. First it was fixtures, over which they claimed copyright and demanded extortionate licensing fees (see example: http://www.bsad.org/0506/repor...). This claim was ruled invalid when tested in court, where in a rare outbreak of common sense it was ruled that fixtures have insufficient creative input to be copyrightable.
The main reason they're going after goal clips so much is that they sell rights to such clips to the Murdoch media. Rupert is using it as one of the carrots to drive subscriptions to his paywalled titles The Sun and The Times. In ostrich-like behaviour the music industry would be proud of, the likes of Vine and 101greatgoals.com are viewed as the enemy, when many brands would kill to get their branding spread so widely in that fashion. Sometimes their aggressive approach to takedowns is at odds with the football clubs themselves. A friend of mine uploaded a clip of goal celebrations from a match he was at to Youtube. The club in question evidently liked it, since they linked to it from their official social media accounts. That, it seems, brought it to the attention of Football DataCo and prompted a copyright claim. His Youtube account was summarily suspended with no appeal, and has never been recovered. This, for a cameraphone clip consisting mostly of crowd scenes.
Then there is the ongoing legal battles over pubs showing games "illicitly". In a manner analogous to the TV blackouts in the US, no match starting at 3pm on Saturday (the traditional time from the pre-TV era) is allowed to be broadcast on UK TV. Since channels covering half the rest of the globe broadcast these matches, some enterprising publicans buy foreign satellite systems, and show the matches to their clientele. The legal situation for this is murky. Purchase of systems from the rest of EU is perfectly legal. The PL cannot claim copyright over the action itself, since it is not its intellectual creation. It instead pursues copyright claims on the grounds of the surrounding branding. (http://www.bbc.co.uk/news/business-17150054) The legal battles will continue, but as with the music industry, the ordinary fan lacks the financial clout to fight a legal battle, no matter how strong the case for the defence may be.