A penguin! FOOL! Better idea, an ANNOYING penguin! FOOL! Better idea, an ANNOYING penguin with a TOP HAT! FOOL! Better idea, an ANNOYING penguin with a TOP HAT and MONOCLE! FOOL! Better idea, an ANNOYING penguin with a TOP HAT, MONOCLE, and a CANE! FOOL! Better idea, an ANNOYING penguin with a TOP HAP, MONOCLE, and CANE with the habit of breaking into SONG! FOOL! Better idea, an ANNOYING penguin with a TOP HAP, MONOCLE, and CANE with the habit of breaking into SONG with a strict 1000 TASK schedule!
Or I always thought a good detective arch-nemesis would be a corrupt police chief (or high official who can trip up the detective using his position).