Safe specifically refers to memory safety, it makes no promises outside of that. By attempting to solve the memory safety issue they also solved things like data races and issues with concurrency. That's all the "Safe" marketing is referring to.
As for stability I've only been using (non-nightly Rust) for a few months now, but haven't come across a single issue. That being said Rust 2018 got a huge overhaul and now that they got a foundation I'm sure there's going to be another roadbump to align to corporate interests but I'm pretty sure after that the fact that companies like Amazon and Microsoft want to use Rust to replace all of their back-end infrastructure leads me to believe there won't really be rocking the boat much after that.