Korrasami (Korra and Asami Sato). This article written by the creators of avatar proves Korrasami is canon.
Moments in the ShowEdit
Characterization in ShipEdit
Varies quite a bit depending on the shipper. Many fanworks have shown Korra and Asami developing a better friendship first before becoming involved romantically. Sometimes hurt-confort is involved, such works where Korra tries comfort Asami after Asami finds out that her father is an Equalist. This also applies to works where Korra's bending loss lasts longer than it did in the Book 1 Finale and Asami is there for Korra. Additionally there are works involving this ship that go into more depth with the conflict with the Equalists.
Like most popular ships, there's at least some crack and humor fics and fanart out there.