The Copenhagen interpretation is (roughly) that the act of perception or measurement (or some other interfacing event) collapses a spread of 'possible' into a single 'actual' (it is essentially destructive, or ever-simplifying). The Many-worlds interpretation is that every possible outcome exists in its own "world" (more accurately universe) via the mechanism of quantum decoherence, instead of wave-function collapse. It is essentially creative or ever-complicating.
Four SF novels that I like deal with these issues in four different ways. Interestingly all four also raise the issue of Aliens, which are Alien not only in planetary origin but in their relation to decoherence/ wave-collapse. There are spoilers but these books have been out for many years.
The Lathe of Heaven by Ursula Le Guin is about a man, George Orr, whose dreams come true, changing the world. It is not clear whether he is collapsing probable universes, or moving himself and his companions between parallel universes (is he a Copenhagen or Many Worlds dreamer?) The Aliens come in at the end, and it turns out they live happily among the various possible universes, never collapsing them. They teach George Orr to live like them. Hence I think the book plumps for the many-worlds interpretation, though not unambiguously.
Quarantine by Greg Egan takes the Copenhagen route. In this story human beings are the only species whose perceptions collapse the wave-function. And this is disastrous for the Aliens, who exist as probabilistic clouds of being. The human act of perception kills them, and collapses the probability waves in which they live. As a response they 'quarantine' the solar system, enclosing us in a bubble so we can't perceive (and hence destroy) the multi-verse.
Stone by Adam Roberts also takes the Copenhagen route, with a twist. In this story human beings can never meet Aliens, because they have collapsed the Universe in a different way, with their alien perceptions. Roberts kind of combines the two interpretations: perception collapses, but each perception collapses in a different way, creating multiple universes. I think he doesn't take this to its logical conclusion, which would have every perceiving individual (not just every intelligent species) isolated in his own collapsing universe.
Hmmm, I said four books and I've forgotten the fourth one I was going to mention. There are plenty of multiple-universe stories (for instance The Female Man) which use these theories of course, but I'm thinking of books which actually grapple with them. Suggestions welcome.