The first comes when the main character, disgraced reporter Mikael Blomkvist, takes time off from his latest assignment to go see a movie for a change of pace:
In the evening he went to the cinema to see The Lord of the Rings,
which he had never before had time to see. He thought that orcs,
unlike human beings, were simple and uncomplicated creatures.
--Chapter 11, page 169
This seems just part of the contemporary grounding of the novel, which is full of references to current events. However, the next goes a bit beyond that. At one point Blomkvist is suddenly confronted by one of his new neighbors, an unpleasant hermit, whose advent is described, apparently from Blomkvist's point of view, thusly:
Gollum had emerged from his cage.
--Chapter 17, page 247
That's the lot so far (I'll add an update at the bottom of this post if I come across any more): another indication of Tolkien's penetration not just into our (American/English/New Zealand/English-language) culture but indeed well beyond it.