Runic writing stopped with the Christianization of Scandinavia. It did however take longer than most people know. The medieval runic alphabet was used as late as the 1500's. The Codex Runicus (1300-ish) for example was written entirely in Medieval runes and is one of the oldest examples of scandinavian legal documents in existence.

Even after runic writing had been replaced entirely by the latin alphabet the practice of writing with runes persisted for decorative purposes, mainly dalecarlian runes which gradually evolved from medieval runes into an increasinly latinized alphabet. Dalecarlian runes were used to mark ownership, make rune staffs (a kind of calendar) and for decorative purposes as late as early 20th century in the Älvdalen region of Sweden (though there is some debate as to whether or not this was an unbroken tradition or not).

You can still find dalecarlian runes on houses in the Älvdalen region and you can still find people who can read and write dalecarlian runes although these days it's mostly for souvenir crafting.