Each player can't cast more than one noncreature spell each turn.
Players may cast any number of creature spells plus one noncreature spell each turn.
Deafening Silence will look at the entire turn to see if a player has cast a noncreature spell yet that turn, even if Deafening Silence wasn’t on the battlefield when that spell was cast. Notably, you can’t cast Deafening Silence and then cast another noncreature spell that turn.
If you cast a noncreature spell that was countered, you can’t cast another noncreature spell that turn.