This, from the excellent Edmodo Blog, shows teachers and students at Meadowbrook High School in Richmond, Virginia, making use of Edmodo in a number of interesting ways. Using Edmodo as a book discussion group is an idea I've found to work particularly well.
From the same source, how Edmodo teachers can save 160 million sheets of paper a week (!!!).
On TechLearning.com, Lisa Nielsen compares Edmodo to Facebook and makes the point that "Facebook is the platform where our students and parents are already communicating".
My experience of using Edmodo is that there certainly always seems to be a learner who eventually puts their hand up and says "Couldn't we do this on Facebook instead?" but, more than anything else, it's Edmodo's greater (ie. total!) privacy that makes it my preferred social learning platform.