New Guide Added for Flash1 Formatter v2 that is created by Team t0c.
This can be sticky if the guide I'm writing is supportive.
I will give you a tutorial on how to format flash1 using Team t0c's homebrew program.
First download Team t0c's Flash1 Formatter v2 which you can download in this thread LINK
Next you will need to extract the program into your GAME folder on your psp through USB MODE on your PSP.
If you don't have something to extract the program then go and download WinRAR on the google search engine.
Once that has been extracted, turn off USB MODE and hold the POWER button untill the PSP Fully shutsdown.
Next after your PSP is turned off, turn it back on again while holding down the R Button. You'll now be in the Recovery Menu.
Scroll down untill you are highlighted on 'Run program at /PSP/GAME/RECOVERY/EBOOT.PBP' this will now load up Team t0c's program.
For those who don't read the README Files then I will tell you. Hold down the buttons Triangle + Circle + Cross at the same time to unlock the program.
Now you have a selections in the menu to choose from. If you're flash1 is corrupted then do a complete clean format. Choose the first selection 'Press Cross to format Flash1' while holding the L button down so it won't make a dump of your bad format of flash1.
After it's complete you man now press the Cross button and it will restart your PSP back to normal.
There you have it, you're PSP is fully restored normally and you can reset your settings to make new settings.
Old Tutorial - Ingore it as it has some false to it, but only use this if Flash Formatter v2 will freeze when trying to format...
WARNING: I am not held responsible if you try and do this stunt on your Flash0 as you WILL brick it by doing this trick I will teach you!

I know what you're probably saying, you should use Flash1 Formatter V2...well I did and it never worked for my psp as it would freeze since I too had this problem recently. I, for one, found another way around this. Everyone probably knows how to format their hard drive right? Well its about doing the same thing as making your PSP as an HDD USB drive to format the normal way, but requires some computer knowledge.
If you too have the same problem when trying to use Flash1 Formatter V2 and freezes when formating then here's a little trick I found out.
First, you go into Recovery Mode by having your PSP fully turned off and then turn it back on holding down the R button.
Next, go to Advance > Flash1 USB Mode. Make sure your PSP is connected to your computer. (You will need your memory stick still inserted as for some reason doesn't show up without it as it is suppose to trick the computer)
Next you need to go to START > Programs > Accessories > Click 'Command Prompt' This will bring up the good old DOS style command GUI.
Ok so now your Command Prompt is opened, but first we need to know your PSP's Drive letter as this is important. You can check it by going to My Computer and it'll show what drive letter your PSP and remember that Letter.
Now lets head on to the Command Prompt. On the program, type this without quotes please... 'Format F:' (F: is my drive letter so ignore this as you need your drive letter for this process.)
Now it asks about another process running at the same time and will ask for you to dismount it and say Y/N? Type 'Y' as it will dismount the pretend memory stick and will start formatting your Flash1!
NOTE:Depending on the size of your memory stick it will take some time to format as it isn't accessing your memory stick but your PSP's Flash1. Also make sure your PSP is pluged in as you don't want to screw up if your PSP is on low battery power and any of the sort.

Once the format is complete, disable your psp's Flash1 USB mode and now exit your psp to boot to your normal PSP mode.
You're probably going to get the BSOD *Blue Screen of Death* as this is normal. As always keep your PSP plugged in the power and now press O for it to...thats right...reformat the PSP's Flash1 and it will reboot back to normal asking you for your new settings.
And there you have it, no more corrupted Flash1!
EDIT: I've added a tutorial for Flash1 Formatter v2 since I know someone wants a guide for it also...