A quashed offence is an offence removed by a Jagex Moderator for various reasons. They were invisible until the Offence System Update; now they are crossed out. There is never a reason posted for the quash, but it is always one of the three following reasons. These were posted in the Official RuneScape Forums by Mod Stevew:
Once the offence is quashed, it no longer counts a strike on your RuneScape account.