Mute, the Silent One is an ability power caster who excels and bursting down crippled enemies and disrrupting enemies during teamfights.
Say No More allows Mute to counter-harass the enemy during the earlygame as it can be followed up by a quick Hush or Punishment. The magic damage associated with Say No More is almost a joke. The one second silence has to be capitalized on. Also, I doubt that anyone that's not fed maybe except an assasin or mage, could deal 10% of you health in 1 second.
Hush is Mute's core burst ability. It should be used to follow up one of his other silences to benefit from the double damage. Cooldown increased and Silence Removed. Also note it's range is somewhat short, and you would have to get in close.
The delay on Null Field plus it's small radius makes it an unreliable silence in the earlygame. However, positioned right, the field can force enemy carries to be in less favorable positions to deal damage to your team. Note that the field is tiny.
Punishment has great synergy with Second of Silence. The ability has a global range, and can be used to pick up assists without even leaving your lane. Increased cooldown so you can't spam this. It allows for counterplay by giving enemies a window of opportunity while its on cooldown.
Mute's ultimate is a global silence great for disruption in teamfights. Note that vision is required to silence an enemy champion. A well timed nocturne ult can negate Second of Silence, but walking out of his line of vision or into a bush can work as well. Also note that the mark won't apply on untargetable enemies or enemies with a spellshield.