The phrase "dig at" means to make a pointed, critical comment or a slight insult. There are several other synonyms that can be used to describe this kind of remark. Some examples include to jab, to mock, to jibe, to tease, to ridicule, to scorn, to deride, to jeer, and to taunt. These words all involve making a negative comment or gesture towards someone, often in a sneering or sarcastic tone. While they may have slightly different nuances, they all convey the same basic idea of taking a jab at someone or something.