Publicly for 1.76 there is a WebKit exploit in the browser and 2 kernel exploits. The kernel exploits work a few versions above 1.76 but there is no public WebKit exploit to use them with.
If you are on 1.76 you can boot Linux right now without needing to compile or host anything yourself; it's as simple as placing 2 files on a FAT32 USB and visiting the live demo of PS4-playground.
No developers that I've ever spoken to have shown interest in running pirated games, so I wouldn't expect this to happen for a fairly long time.
For later firmwares there is little you can do unless you are a developer. You can run ROP chains in web apps which are statically compiled to outdated versions of WebKit, which is enough to test whether they are vulnerable to most kernel exploits, and you can do a few other things. These apps don't have permission to use the JIT system calls so they can't be used for full code execution unless you want to trigger a full kernel exploit with just ROP which is pretty hardcore (very difficult to do multi threading for example).
There are plenty of WebKit bugs which could be used to exploit the browser on the latest firmware for arbitrary code execution (
http://webkitgtk.org/security/WSA-2016-0002.html http://webkitgtk.org/security/WSA-2016-0003.html) however the WebKit team won't publicly publish the details of these bugs, but they could be found by manually going through thousands of WebKit commits if you wanted.
Generally I wouldn't expect anything for latest firmware to be released until it has been patched though.
In terms of developers to follow, there's myself (
http://twitter.com/CTurtE), who was first to publicly announce both kernel exploits, and I've published articles documenting them on my site (
http://cturt.github.io/ps4-3.html and
http://cturt.github.io/dlclose-overflow.html). But I don't have much interest in the PS4 at the moment. There's also kR105 (
http://twitter.com/kr105rlz) who wrote most of the public Linux boot loader. Most other developers tend to prefer to work privately, so I won't add them here, but I've found wololo.net to be by far the most accurate site for PS4 hacking news which you should follow if you don't want to miss anything.
I hope this answered all of your questions.
