When started a new project, particularly one that you have to clean up frequently, use a plastic bin to collect all the parts you need. Clear plastic bins are great for this. For sewing projects, I like a bin that is approximately 16 x 11 inches and several inches deep with a lid. For my origami, I used a large, flat tote so I can spread out the folded flowers.
If your project is a costume or for occasional use, the bin can double as convenient storage as well. When doubling up on usage, be sure to buy bins of similar size for easier stacking.