At the sake of taking a departure from usual well thought-out and intelligent commentary of this forum, the answer to the question is yes, all wrestlers are required to wear a proper undergarment under their singlets. Failure to do so will result in the same penalties as taking the mat without headgear, taped laces, etc. Some equipment companies are offering lycra undergarments now that are supposed to be much more comforable than regular underwear.
Hope this answers your question.