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.
15$\begingroup$ I think you can put just about anything on your profile, provided it's not offensive. $\endgroup$– Todd Trimble ModSep 27, 2016 at 19:21
1$\begingroup$ @Todd: I think you can run into trouble with some non-offensive things as well -- e.g. if you violate copyright or disclose secrets. $\endgroup$– Stefan Kohl ModSep 27, 2016 at 20:03
22$\begingroup$ Well, then expand the meaning of "offensive" to include cases where one has committed an offense. :-) $\endgroup$– Todd Trimble ModSep 27, 2016 at 20:16
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.
6$\begingroup$ Thanks for the answer. Concerning the last paragraph, I believe this is true if you apply as a post-doc or a non-tenure assistant professor. These are positions that are almost always advertised on mathjobs, and if someone in a university knows you from MO, he/she will probably recognize you anyway when you apply. But for positions that are not widely advertised or not even existent before you apply, I believe that MO can be a nice and soft way to throw a bottle to the sea, with the message "I am unhappy where I am. Please hire me!" $\endgroup$– JoëlOct 4, 2016 at 0:20