In The Gathering Storm, Rand is sporting a rad "new" sword. "He fingered the cloth-tied hilt. The weapon was long, slightly curved, and the lacquered scabbard was painted with a long, sinuous dragon of red and gold. It looked as if it had been designed specifically for Rand-- and yet it was centuries old, unearthed only recently." [TGS 1: Tears From Steel] At a signing, Brandon Sanderson further tantalized us by saying it was discovered, "In water, under a statue, not near Falme"
So what sword is it?
It's probably Artur Hawkwing's sword "Justice". Rand says he recognizes it from his own memories, so that rules out it being a relic from the Age of Legends. Lews Therin doesn't recognize it as his. And the only time we know Rand might've seen a very old sword would have been the Battle at Falme when Mat used the Horn to bring back Artur Hawkwing and the rest of the Heroes of the Horn. Later in the same book, when Rand return's to Falme, he muses again, "..[the sword] made him think of Falme."
So we have a very old sword, but not old enough to be from the Age of Legends, that reminds Rand of Falme, even though it's not from there. It's probably Justice.
Another interesting tidbit about the sword: It's also Brandon Sanderson's. On one of Brandon's first visits to meet with Harriet and team Jordan about completing the series, Jordan's cousin Wilson offered Brandon his pick from Jordan's rather extensive collection of weaponry. Brandon chose a black sword with a red dragon handpainted on it. It's now on display in his home. You can view pictures of it on Brandon's blog.