Broken points also have character... could be an impact break, could be due to cold weather, could be someone stepped on the dern thing... I keep them all. I always hold out hope that I might find the other part. They're in terra cotta flower pots in my dining room. But take into consideration too, that broken points of a certain age or design (or age and design) are-- or can be-- worth quite a chunk of change. A paleo broken point can be worth a lot- more than a crappy little worn out knife w/ no identifiable base. That said, in the beginning- when I first started out in this 'hobby' (chuckle, chuckle), I kept everything- first on the window-sill, then in a frame. It depends on the piece. If it has a really cool base, I'll keep it in a location frame. If not, I'll put it in a flower pot. Every now and then I go through my flower pots and find things I didn't understand when I found them. As you go, you tend to learn a bit, so... there's that.
I pick up flakes too, if they're pretty- and of a unique lithic. One day- probably now- I have enough to fill the top of a box- style coffee table. Nice w/ dim light shining from below- or some artistic endeavor. (but I should say-- as far as flakes go, opinions vary. I'm of the opinion that you should leave what you don't need. If you have enough 'evidence' for your own self, leave the rest, so someone else can see- and look for artifacts later, when you're dead and gone. Initially, my enthusiasm was such that I'd scour and clean out everything I found. Now, years later, I don't do that. I'll even leave perfectly good chuncker/choppers so the next generation has a hint, and continue the hunt (and research). Good question- thanks for posting! Yakker