Baghdad has not been safe since the overthrow of Saddam Hussein. It has been over a decade since this happened and things are no more stable there now than they were then. With all the recent incidents of mass murder in various countries, I don't think there really is anywhere that is safe 100% of the time. Some areas seem to be more prone to incidents than others, Iraq, France and Turkey are a few examples.
No, I do not believe that Baghdad is a safe place. Baghdad is an often targeted city and including the attack by a suicide bomber that killed 22 it is evident that a threat still exists in this area. It would be unwise for anyone to choose to travel here.
I do not believe Baghdad is safe and would never choose to visit personally, based on this fact. I do not believe that anywhere is completely safe, but based on the number of deaths, injuries and recent events in the area I would say that people should avoid it if possible.
Baghdad has endured conflict for an entire generation. This dates all the way back to the Gulf War, which was approximately 20 years ago. It is too easy to walk into a Baghdad market and hurt innocent people, especially by suicide bombing. Baghdad does not have a way to police this before it happens.