拎 Informal, something that can be carried away easily, (Cantonese). 拿 A more generic term, used in Mandarin. 提 formal and in Classical Chinese. In modern colloquial, 手提 = cellphone. In Classical Chinese, to carry by hand 手提玫瑰献芳魂

In fact, there is no difference between the three essentially.

拎 and 提 have the same meaning. But if you want to dig deeper, 拎 describes a static status. It means you are holding something (off the ground). 提 refers to the action that lifts something from a lower place to a upper place. Personally I think 提 is used by people from the southern part of China more often, and 提 is used by people from the southern part of China more often. I'm from Shanghai. I never say 提, but I use it in writing. 拿 just means 'carry'. It doesn’t have to be from a lower place to a upper place. If you say拎着/提着/拿着, we all understand, they mean the same to us in verbal Chinese.

拎 着重指提着轻小的物体 more emphasis on small and light objects，跟提区别不大 而 拿 用处很宽泛