Usually a jock strap (with no cup), or underwear, or both.
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I believe you are only supposed to wear compression shorts. Those are the shorts that are very tight and insulated so you don't sweat so much. At my school you are not allowed to wear a cup because it could really harm the other wrestler.
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First they aren't called tights they are called singlets, and the purpose of them is to not have clothes for the opponent to grab on to because the singlet is tight and not baggy