The PS4 has full stack exploit for firmware 5.05 and a private exploit for 5.5x systems. (28/05/2018) It's finally, finally out.
The exploit supposedly works on 5.07 firmwares as well. Do not update from 5.05 to 5.07 as no games require 5.07 to run. If you are on 5.07 continue with any step that says 5.05.