What is the significance to Laura? Why is the unicorn single out? What are its symbolic implications?

The significance of the unicorn to Laura is that it is unique, just as Laura is unique. It also does not quite fit in with the rest of her Glass Menagerie, just as Laura does not fit in with a conventional way of life. All in all, the unicorn is symbolic of Laura; when its horn is broken off, it is no more unique and it gives us the impression that she has no hope of exploring any of her good qualities in the future.