Their highest leader was the Witch-King, who led the Siege upon Minas Tirith. But at Cirith Ungol hundreds of them were killed by their own Mordor orc brethren over Frodo's Mithril shirt, including Gorbag, one of their leaders. Eventually Shagrat brought the Mouth of Sauron the shirt and told the authorities of Barad-dur that some rebel Morgul orcs were on the loose since some escaped the pass of the spider.
Even though Gothmog was the Witch-King's lieutenant, he was a Morannon Orc. Morgul orcs were usually prejudiced by other orcs, mainly the Uruk-hai, mainly "Morgul rats" or "scum". Shagrat was a good example of showing prejudice against them.