Here's one of the things I love about felted fabric... if the fibres are properly matted together, you can cut through the fabric with a pair of scissors and it will not fray. Take a close look at the scraps pictured below, and you will see what I mean.
I don't ever throw away my felted scraps... even the tiniest ones come in handy for embellishing other projects. If you honestly feel you have no use for them, you might think about donating them to your local kinder or school art department.
Here's an example of one of my projects where those tiny scraps came in handy. The crest, beak, wings and feet are all cut from felted scraps.