The title says it all. I do not see why it should be a problem, but I realize I have never seen anyone doing that here (or I forgot), so I ask.
It shouldn't be a problem: you are unlikely to offend anyone by doing so, and it is unlikely to be considered as advertisement or spam.
If you put that you are looking for a job while currently holding a job ... that's your problem with your employer, and not a problem with MathOverflow.
The likely reason that you have never seen anyone doing it here is that most people, I would venture, hold little optimism that advertising one's availability on MathOverflow will lead to any additional job prospects.