The fastest way to start searching for vessels (or ports, lights etc) or to track down certain ones that interest you, is by using the appropriate Search field on the MarineTraffic main menu.
Notice that while you are typing your search term, a drop-down list will come out containing vessels that match your search term.
Clicking on the name of the vessel that you are looking for, will result in getting you to her Vessel Details page where you will be able to view all the related information.
Notice the "Show All Results" option at the bottom of the drop-down list.
Clicking on this option instead of on the name of a vessel will result in opening a Search Results page. All vessels that match your search criteria will be displayed in it.
Choose the vessel that interests you from the list to get to her Vessel Details page.
Note that this type of search is the only one where you may get search results regarding Ex-Names of vessels - click on the vessel's ex-name to get to the Vessel Details page of the vessel under her current name!