Let's discuss "zero". There is one school of thought that believes that every number except one is plural. Adherents of this school of thought would say "zero degrees". I am not one of these. I and many others believe that anything larger than 1 and smaller than -1 is plural. Zero, we believe, is singular. Some examples of the established usage of zero as a singular quantity are: 1. The Bank of England is at present offering a zero rate of interest. 2. We shall set three o'clock as zero hour for our air strikes on the enemy. At the end of the day, it is a matter of personal belief and personal logic, and don't let any friends or teachers bully you into picking one or the other without understanding and articulating both points of views and thinking about them yourself.

Then answer is "zero degrees," but I can't say I really can give a logical explanation. The only time you use "degree" is when the number of degrees is PRECISELY 1. Zero degrees, 0.5 degrees, 1.1 degrees, one-and-a-half degrees... No, that's not quite true. "0.1 degrees," but "one tenth of a degree." There are situations in which "degree" means hierarchical levels of something. In this case, "degree" is always singular. He is a 30th degree Mason. He has first degree burns. The jury returned a verdict of murder in the second degree. In mathematics and the sciences, one can continue the ordinal numbers down to zero, and "zeroth" is a good dictionary word, so you could speak of a "zeroth degree polynomial," and Google is finding me references to "zero degree polynomials." I think it would be better usage to speak of a "polynomial of degree 0."

I wonder in what circumstances one would say something like "The temperatures haven't dropped to zero degree." Smelval, where did you see this in the first place? For a learner, or even a native speaker, it is important to learn to use words and tenses in the right context. As far as grammar is concerned, in the sentence put forward by Smelval, "temperatures" could be right if one was thinking of multiple temperature readings over a number of hours or days or months. Zero is not plural, and so we would say "zero degree". I do hope that Smelval will continue this interaction by discussing the context with us.

Think of it like this. Unless you make it singlular with "a/an" than it's plural. Such as, it's 90 degrees out today VS it's a 90 degree day. When it comes to 0 degrees the same rule applies unless you're saying It is a 0 degree day. Hope that helps.

The temperature hasn't dropped to zero degrees.