What goes on under your singlet is important. The singlet is a standard competition uniform for collegiate, freestyle and Greco-Roman wrestling. It consists of a single piece of stretch fabric that covers the upper legs and torso, much resembling the one-piece bathing suits of the late 1800 and early 1900s. But your singlet is not generally enough to remain comfortable and appropriate during competition.
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Wearing just one layer of stretch fabric to competition leaves very little to the imagination. To avoid embarrassing themselves and others, wrestlers usually wear some kind of undergarment -- most frequently underpants or a jock strap. Female wrestlers will also wear a T-shirt or sports bra for the same reasons.
Wrestling is not only a contact sport -- it's a combat sport. This means the potential for injury is higher than in many other activities. Protecting the softest parts of your body is a priority. For men, a groin cup is a common addition to the singlet uniform -- though since their advent in the early 2000s, many wrestlers prefer to simply wear compression shorts. Women will sometimes wear female groin protection, but this is less common than for males. Women do frequently put on a thicker sports bra to avoid discomfort in the chest, but rigid breast protection cups are too uncomfortable to wear while grappling.
Hurt wrestlers sometimes compete while injured. In some cases, this requires them to wear special gear underneath their singlet. A minor groin injury might result in an athlete wearing a groin cup when he would not otherwise. It's also common to find tape or other compression underneath a singlet, stabilizing muscles that have sustained minor or moderate dings throughout the season. A wrestler with a skin rash -- if permitted to wrestle at all -- may compete in a T-shirt to minimize skin-to-skin contact with his opponent.
Follow the Rules
A final reason to wear something under a singlet is related to uniform standards for a team or a league. Some teams may ask wrestlers to wear compression shorts or a team T-shirt beneath the uniform to create a cohesive look for the entire team -- especially if they have female participants who must wear a shirt. Although rare, a wrestling league may require specific undergarments of all wrestlers who participate in sanctioned competitions. This is usually related to modesty concerns.