iOS 5.01 unlock/jailbreak and preservation of baseband.

1. Redsn0w (win/mac). +preserves installed software, all apps is kept on device after jailbreak.

2. Ph0nagetool. Jailbreak/modifies iOS file on computer, then uploads to device. +keeps baseband, so unlock procedures is available for locked phones. All previously downloaded apps is replaced by uploaded/modified iOS file.

If Cydia doesn't work after redsnow (jailbreak), run Redsnow again with other options (reinstall cydia). White icon common, boot tethered choice from cydia might be a remedy.