The phrase BSOD has been used for a number of different problems by various people. The screen is Black and the device is unresponsive.
There have been a number of different remedies reported and that is because there are a number of different problems. These remedies were ...
If you are using a charger or USB cable with the Bionic that did not come with it ... stop. Use the charger and USB cable that came with the Bionic. There is no way to know what damage may have been introduced by using third-party hardware to charge your Bionic. That damage may not have been limited to just the battery.
Power On the device (it was powered OFF).
Fully charge the battery that is at 0%.
The validity of "conditioning" the battery is open to debate. Some claim that it is worthless. Others swear that it is beneficial. You could try it. Charge to 100%, run it down to 0%, repeat three times. (I always "condition" new batteries.)
Reset retaining your customizations (hold down Volume UP/Volume Down/Power buttons until the device reloads. This is the same as opening the back, pulling the battery, replacing the battery, and powering ON.
Pull the back, pull the battery, make sure both cards are properly seated and not off at an angle.
Go back to where you bought the device and test the battery in another device ...
-- If bad ... replace the battery (standard or extended) with a new battery.
-- If good ... replace the SIM card.
If it still fails do a factory reset. This will wipe all your customizations and installed Apps.
If it still fails replace the device.
... Thom