How much do prior publications matter for PhD admissions?

A little background about myself:

I graduated from my undergrad with a 3.7 GPA and currently I'm doing a master of laws, on track to graduate with similar grades. I'm really enjoying the independent research aspect of my masters, so I intend to apply for PhD programs this coming application cycle, primarily in the US and Canada.

I have heard from professors before and have been encouraged to submit the papers I wrote for coursework to academic journals, but at the same time, I was warned how selective the process can be to ultimately be selected for publication. I have been told that already having a publication in my name will open many doors in terms of acceptances, but I am wondering if anyone has any insight on how much prior publications matter to be admitted into top PhD programs in law?