Answer:
Yes.  However, best results will often be obtained by using shotguns designed for the given purpose.  Field, skeet and sporting clays have a similar point of impact; targets can be on the rise, crossing or dropping.  Trap however always has targets on the rise so trap shotguns have a higher point of impact.  Additionally, there are certain features on the different types of shotguns that benefit each of these sports.  For more information see the Firearms section of the website.