Update: It's now been reported that the JAP version of Lumines does not work with the exploit, so those interested should seek out a USA/EUR version for the time being. Also, the exploit doesn't seem to work properly if your PSP is set to non-English settings. It's likely something to do with unicode when displaying the profile name, but if you are having problems (especially with the EUR Lumines) then try switching your PSP to 'English' settings and enjoy.

Today the Illuminati Exploit For PSP 3.10-3.50 is available, which includes a "Hello World" for PSP Firmware up to v3.50 that opens the door to homebrew, a v3.50 downgrader, and more! To quote from the ReadMe file:
The first working homebrew on all firmware.. including firmware 3.50. Based on Lumines, the "Illuminati" exploit is a user-mode exploit using a buffer overflow in the savedata file - similar to the GTA exploit.

So if you've got a copy of Lumines (the UMD disc), then you can run homebrew on your PSP, no matter what the firmware version is. Currently, the only homebrew compatible with 3.50 is this Hello World demo, but in the future a new HEN and downgrader will be available!

To make the exploit work:

* Check that you have an EU or US version of Lumines (ULES00043 or ULUS10002). Currently we don't have a version for the Japanese version, but we're working on it, as soon as we get hold of a Japanese UMD.

* Extract the contents of the 'MS_ROOT' folder from the RAR file into the top-level of your memory stick. (HINT: If you wind up with an 'MS_ROOT' folder on the stick, you've done it wrong).

* Start the game, and as soon as it gets to the 'Press START' screen, press START. If you wait until the demo has started, the exploit may not work.

* The screen should go blank, and the exploit will start after a few moments. It doesn't do anything except look pretty at this stage - but feel free to enjoy the first homebrew on your v3.10 - v3.50 PSP!
PSP Files: Random Topmenu PRX Mod v2 For Custom XMB For PSP, Hybrid Firmware 3.00 IE-A Update 1 For PSP, and PSP-PDA v1.3.1 For PSP.