I am not a woman. It is very important that women get a properly fitted sports bra, this helps prevents damage to the breast region. And avoids discomfort that may occur.
It is very important that both sexes have very comfortable underwear to avoid chafing. You can’t wear any garment twice in a row. I would not wear old underwear either because the elastic may have become hard and could also cause chafing. Buy some anti-chafing cream to avoid painful and embarrassing problems.
When using running tights or bike shorts I would even consider not wearing any underwear. Because the underpants could rub against the tights and bike shorts and could cause some chafing.
So being prepared for your training is important.