People who begin to abandon something or someone that is failing or about to fail with great haste and having only personal wellbeing in mind. The phrase can also be used in the singular, but this is less common. After the new party leader's embarrassing and incendiary comments, many have begun distancing themselves from him and the party's ethos, giving the impression of rats deserting a sinking ship. It's true that I am resigning, but I am not a rat deserting a sinking ship: I have full confidence in this company, I am merely changing course in my career.