You can do both depending on the style of bag it is. A “handbag” is on that has handles or not that can only be carried. A bag that has either a shoulder strap or a “crossbody” strap can be worn.

In standard English grammar, you typically carry a handbag, whereas you wear a backpack or jewelry. The distinction lies in how the item is supported: "wear" is used for items kept on the body (like a hat, watch, or clothing), while "carry" is used for items held or supported by the hand or arm during transport. However, the line has blurred with cross-body bags, where it is increasingly acceptable to say you are "wearing" the bag because it is strapped across the torso. In fashion contexts, "wearing" can also refer to the entire ensemble (e.g., "She is wearing a Gucci handbag"). Generally, if the bag is intended to be held in the hand, use "carry"; if it is strapped to the body for an extended period, "wear" is colloquially common.