It's not that .Net Core itself is unstable or anything, it's just that the ecosystem of packages and libraries is fairly immature, and there is still a lot of mismatches in support and dependencies.

For example, if you use core, you can't use entity framework 6, you have to use EF core, which has most features and if you are building a new app you'll probably find a way to do most things you want, but if you are porting something you might struggle. And that means you can't use anything that is built on EF 6 either. The dependacy chains seem to be muddled. One library will seem to be fine on core, but one of it's dependencies will only work on the full .net fx.

Another problem I hit was with doing some dynamic code generation. Roslyn is supported, but code dom isn't. That would be fine if I was starting from scratch, but I had a bunch of useful existing that that all depended on code dom that would have been a lot of extra effort if I couldn't reuse.

I would say if you are starting a new project and it's a fairly standard website you'll almost certainly be fine. But be ready for a lot of pain if you have any code to port.