There's a lot of variation from country to country, and in the United States, from one state to the next. Even in a single state, the rules can change depending on other factors, like when the dispute arises and whether the parents are married.
Unsurprisingly, the laws here in the United States tend to favor the use of the father's name, but again, these can be very fact-specific. There was a law review article several years back summarizing the interstate variations in child naming laws that might be helpful: Naming Baby: The Constitutional Dimensions of Parental Naming Rights, 80 Geo. Wash. L. Rev. 159 (2011).Naming Baby: The Constitutional Dimensions of Parental Naming Rights, 80 Geo. Wash. L. Rev. 159 (2011).