Key:
Homebrew Available
No Homebrew Available
DAX Firmware

Sony Firmware

Homebrew Available
1.00- All but a few homebrew available

1.50- All homebrew available

1.51- No homebrew available-Upgrade to 2.00-2.71

1.52- No homebrew available-Upgrade to 2.00-2.71

2.00- User mode homebrew available
Non TA-082- Downgrader available
TA-082 users can update to 2.71 and emulate 1.50

2.01- User mode homebrew available
Non TA-082- Downgrader available
TA-082 users can update to 2.71 and emulate 1.50

2.50- User mode homebrew available
Non TA-082- Downgrader available
TA-082 users can update to 2.71 and emulate 1.50

2.60- User mode homebrew available
Non TA-082- Downgrader available
TA-082 users can update to 2.71 and emulate 1.50

2.70- User mode homebrew available
Non TA-082- Downgrader available
TA-082 users can update to 2.71 and emulate 1.50

2.71- User mode homebrew available
Non TA-082- Downgrader available
TA-082 users can emulate 1.50

2.80- User mode homebrew available-Kernel access found (Downgrader for Non TA-082 coming soon)

No Homebrew Available
2.81- No homebrew available-If exploit is found kernel access will be available

2.82- No homebrew available-If exploit is found kernel access will be available

3.00- No homebrew available-If exploit is found kernel access will be available

3.01- No homebrew available-If exploit is found kernel access will be available

3.02- No homebrew available-If exploit is found kernel access will be available

3.03- No homebrew available-If exploit is found kernel access will be available

DAX Firmware
(Can only be installed from 1.50)

2.71 Special Edition

Features:
Launch 1.00, 1.50 & 2.71 homebrew.
Launch ISOs directly from the XMB.
Recovery Menu.
Ability to load .prx modules.

3.02 Open Edition

Features:
Launch 1.00, 1.50 & 3.02 homebrew.
Launch ISOs directly from the XMB.
Launch any PS1 game downloaded from the Playstation Network on any PSP.
Launch UMD Movies from any region on any PSP.
PSP Region Selector.
O/X Main button selector.
WMA & Flash Enabling without internet access.
Recovery Menu.
Ability to load .prx modules.
Location Free and Korean Language have been removed.

A quick reference guide. Credit goes to the developers, keep up the good work

NOTE: Technically 1.51 and 1.52 do not have homebrew, but I still count them as homebrew compatible considering people can get to homebrew compatible firmwares from them.