I own an RX480/580 and play at 1080p with freesync at 144Hz. Many games struggle to hit 60Hz, while many are 60Hz+. What's nice with the 580 is that it keeps the minimum frame rates above 30 so you don't get the occasional lag.
A 560 is better suited to 720p, but will work at 1080p up to 60Hz if you reduce the settings a bit. You will definitely have to reduce the settings in many games to get the minimum frame rates up to reduce lag.
I consider the AMD 580 or NVidia 1060 the minimum card for gaming at 1080p. Too bad they are so expensive now. I purchased my 480 8GB overclocked for under $400 Canadian and the 580 is now up to $600+. The best gaming cards where always in the $600 territory, now the worst gaming card is that...