What's the reason for the switch ?
For us ( the company where I work ) it would be very difficult to rewrite our applications to core, so we decided to only introduce core to new in-house projects.
We're pushing for it primarily for cross-platform reasons. Currently our developers have the choice between MacBooks and Windows machines. .NET developers who choose MacBooks need to run Windows in a virtual machine which can be painful to do constantly. In addition, our application servers are currently Windows which means higher costs due to licensing.
We're moving to .NET Core and basically moving into the Docker ecosystem so that developers can run the operating system of their preference without requiring a virtual machine for development and so that we can use Docker containers on Linux hosts for running the services.
I'd agree that major companies are probably not using .NET Core (aside from Microsoft I'd imagine). As far as it being production ready, I believe it's near or at that point. 1.0 shipped in June. My employer is starting the transition from .NET to .NET Core and we're pretty confident that it's ready. Having said that, we don't have anything currently in production so I don't really have much data to back up that statement.