Why did Singed get himself caught? To lure Warwick to the prison? Then what? Was his plan actually to have it recognize Jinx and calm down? No, because he remarks he never expected that. So his plan was just to what? Die? It only worked out for him because of incredible plot contrivance.

Knowing a bit of Singed lore and his role with Noxus, to me he wanted to show off his creation, a killing machine that he could control, to Noxus to get funds. Also, Warwick attacked the Piltover police instead of attacking Noxus forces, so as not to anger Ambessa.