The term "local law" is a bit confusing in this context since local law is really municipal ordinances, as opposed to state or federal statutes. Nevertheless, no matter what kind of law is involved, as the above response pointed out, it is the majority opinion that has the most immediate effect on the law, locally, statewide, or nationally.
However, it should also be mentioned that there have been many occasions in history when a dissenting opinion became the law of the land subsequently because a dissenting opinion can lay down a path of reasoning that a court is willing to follow the next time around. There have been seismic shifts in the law that were begun with a dissenting opinion.
I think that the majority opinion will usually have the most effect, because this is the opinion that explains the law. The majority opinion means the majority of the justices agreed on the interpretation of the law. The dissenting opinion can be valuable sometimes when the law is very controversial.