The trident was used for hand-to-hand combat and to aid them in their illusions. It can tear into others' illusions with its three spikes. During the Kokuyo Arc, Mukuro used his Trident to create a "pact" between whomever he wished to use his Second Realm on when he was under the influence of the Possession Bullet, but the blades on that one broke and was replaced. However, in the Future during the battle with Glo Xinia, he remembers the moment that Mukuro stabbed his Rain Owl, letting him possess it, which means that he had that replaced the blade.
Chrome uses Mukuro's trident as well. The trident does the same thing that Mukuro's does and can fit into a small bag by retracting the poles of the trident and the handle. Since Chrome met Mukuro and his illusions are keeping her alive, they are bound together by their tridents. If hers or Mukuro's trident breaks, Chrome's illusionary organs would fail; however, the trident can be reconstructed.