In response to your other post, which asks about musicals....
Some opera companies also produce musicals.  So then it depends if AGMA has an agreement with that company.  For example, I was on an AGMA contract when I did "A Little Night Music" but not when I did "Sweeney Todd" or "Porgy & Bess".  These are all rather common in opera repetoire.  Also things like Gilbert & Sullivan, etc. For more information about AGMA, go to 
www.musicalartists.orgOther companies producing musicals may or may not have Equity contracts.
For other performing arts union information, this is pasted from 
http://www.aflcio.org/aboutus/unions/ Associated Actors and Artistes of America (4As) | Contact info 
    Actors' Equity Association (AEA) | Contact info | Website  
    American Federation of Television and Radio Artists (AFTRA) | Contact info | Website  
    American Guild of Musical Artists (AGMA) | Contact info | Website  
    American Guild of Variety Artists (AGVA) | Contact info | Website  
    Hebrew Actors' Union Inc. | Contact info  
    Screen Actors Guild (SAG) | Contact info | Website  
    The Guild of Italian American Actors (GIAA) | Contact info | Website      
For bonus points in class, look up the entire name of IATSE.  It's longer than you think.