I know she's a little more major than most of the other characters I've covered recently, but kevenn requested that I make a post about her, so why not? Besides, she's really only a major protagonist in a few books, although she shows up in the background very frequently.
Basically, the story behind Scraps is that Margolotte, wife of the magician Dr. Pipt, wanted a maid, so she made a girl out of her grandmother's old patchwork quilt. Pipt had the necessary magic to bring her to life and give her brains. Margolotte was careful to give her only the brain qualities that she felt a servant needed, but Ojo added others when the woman wasn't looking. So Scraps ended up being incredibly active and playful, prone to breaking out into nonsense verse, and stubbornly independent to the point that she refused to be a servant at all. After she accompanied Ojo on an adventure, Ozma granted her permission to live at her palace in the Emerald City, and she's done so ever since then. Despite her craziness, she's quite clever in her own way, and has provided some invaluable suggestions to the other celebrities in several books. She and the Scarecrow were immediately attracted to each other, and while L. Frank Baum and his successors never did much with this, there is a book written comparatively recently in which they get married.
When Ruth Plumly Thompson took over the series, she kept Scraps basically in line with Baum's characterization, although some readers have observed that her Patchwork Girl acts somewhat more childish than Baum's. She plays a major role in Gnome King, and shows up with a verse or two in many of the other Thompson volumes. John R. Neill used Scraps pretty frequently as well, but he didn't appear to like her very much, making her somewhat of a belligerent jerk who was always getting in people's ways and trying to box with them. She also hangs around Jack Pumpkinhead more often than the Scarecrow, something that Neill never really explains. Interestingly, it was Neill who gave Scraps her most central role in Runaway, which was unreleased in his lifetime but finally published in an edition illustrated and edited by Eric Shanower.