Most men wear snug lycra cycling shorts without underwear for serious road work. These specialized shorts include a built-in padded chamois in the crotch for cushioning on the saddle. Bikers will lube their groins to prevent chaffing, with products made just for this purpose.
Men's running shorts often include a lightweight liner for support. I've found them ineffective for controlling annoying flopping, so in the past (when I could still jog) I'd cut the liner out and wear a swimmer's jock instead. Same as a regular jock, except the waistband and leg straps are narrower, for less chaffing and a freer feel.
Wrestling singlets seen in HS & college competition are worn without any underclothes.
For myself, when biking long distances I wear non-compression shorts, basically mid-length gym shorts. I've had varicose veins in my calfs since age 30, and compression shorts over my thighs make them worse. It's too hot here to wear full-length lycra tights, but they sell biker's underwear briefs I wear under the gym shorts, that have those chamois pads in the crotch.