What happened to the Death Note? I realize that they (Near and the police) probably hid it somewhere, but after Light dies, would yet another Kira appear? Or would Ryuk take it back? I'm assuming the latter, but when Light dies, Near has the book. Light merely has his scrap paper in his watch. If Near still has it, would it just disappear? Or would Near become the owner?
Every Shinigami has their own Death Note. There were 4 Death Notes in the human world. 2 of Ryuuk's, 1 of Rem and 1 of Rem's friend who died after saving Misa. 

One of Ryuuk's was given to Light at the start of the story.
One from Rem's friend is in Misa's possession.
The other two are in their respective Shinigami's possession.

Later, Rem saved Misa's life by writing Watari and L's name but burned the book so that it won't be used by Light, making the number of Death Note in human world down to 3.

Edit: as mentioned by Arcane in the comment, the book was not burned down in the manga chapter 58. I made mistake here because in the movie it was burned down. Thus there are 4 books by this moment. Will edit the rest of my answer after I finished re-reading the manga.

By the end of the story, Mikami possessed a fake Death Note made by Near and co.
Near possessed the Death Note that used to be Misa's.
L's team (by now Light is L) owns one that used to be Light's.
Ryuuk possessed his own.

Near then burned the two Death Notes after Ryuuk killed Light. Thus no Death Note are in human's possession. This was told in chapter 107 of the manga.