I follow the developments in quantum computing from various sources. My starting point is always the same: John Preskill’s Twitter feed.
The term “quantum supremacy” was first used by John Preskill. He explained his reasoning and concerns in a Quanta Magazine article titled “Why I Called It Quantum Supremacy“.
On October 23, 2019 the Google team headed by John Martinis (Chief Scientist Quantum Hardware) and Sergio Boixo (Chief Scientist Quantum Computing Theory), published a paper in Nature claiming that they achieved quantum supremacy in a specific task. Their announcement can be found in this URL
The scientific paper is open to public at the Nature site (click on the “Download PDF” button): https://www.nature.com/articles/s41586-019-1666-5
For a technical discussion of this quantum supremacy experiment I refer you to Scott Aaronson’s blog post: https://www.scottaaronson.com/blog/?p=4372
New York Times article on the quantum supremacy announcement can be found here.