Developer AlexZzZ recently released a proof of concept exploit for PS4's running version 5.01. This exploit was later independently confirmed to also work on 5.05 by mathieulh. This combined with Qwertyuiop's release of a kernel exploit for 4.55 recently has allowed for a full jailbreak for 4.55 to be created.
This exploit relies on a recently discovered Webkit vulnerability, CVE-2017-7005. This is the same vulnerability which was used by Team Reswitched for the Nintendo Switch. It is possible that the exploit for the PS4 was ported over from the one already created for the Nintendo Switch.
In order to have a fully functional jailbreak, there are two exploits needed. One is a usermode exploit which gives limited code execution control, and the other is a kernel exploit which gives privilege escalation allowing for full access to the system.
With the Webkit exploit as the usermode exploit and Qwertyuiop's kernel exploit of 4.55, there is full access up to firmware 4.55, which allowed for Qwertyuiop to create the 4.55 jailbreak.
Here's a full 4.55 PS4 jailbreak: http://crack.bargains/455/
— qwertyoruiop (@qwertyoruiopz) February 27, 2018
Unfortunately, users on versions 5.0X do not get a jailbreak. They do however get access to the usermode access, but that has historically not been very useful.
As for the PS4 beta firmware of 5.50, it is unclear if the Webkit exploit works for that firmware allowing for a jailbreak. There are two sides to that story with some claiming it does and others saying it does not work. It is also likely that if it does work, it will be patched by the end of the beta. There is not clear confirmation of whether or not that works.
You can get the Webkit exploit here
For users on version 4.55, the jailbreak can be found here
