In Mt 12,38-42 Jesus refuses the demand of Pharisees and scribes for a sign, that would disprove the accusation that his miraculous acts are a result of his alliance with Devil, (Mat 12:27) and that would also unambiguously reveal his identity. Jesus answers the request with a half-ironic reference to the sign of Jonah.
In my exegesis I have concluded, that this half-ironic reference is intended firstly to recall not only the preaching (i.e. the enouncement of judgment and call for repentance) of Jonah, but also his person (contra Luz), and secondly and more importantly, to point to Jesus himself. By the cryptic reference to the sign of Jonah Jesus announces to the Scribes and Pharisees that no other extra sign will be given to them except his own person and ministry - which includes preaching and healing and which they have already witnessed but refused to accept.