Folly is the name of Kvothe's present-day sword. It hangs on a mounting board (made of Roah) in the Waystone Inn. It is most likely not Saicere as the descriptions do not match.
Speculation[]
Some say that Folly may have been Cinder's sword as he carried "a sword that does not reflect light" and Folly is described to be "dull grey-white" and does not show any scratches, almost as it's color isn't a reflection of light. If Kvothe kills Cinder, it is possible he took his sword. Abenthy's letter in Rhetoric and Logic says "Remember your father’s song. Be wary of folly.", since his father's song was about the Chandrian, one could link "folly" to them, but it highly unlikely.