Is barbed wire fencing allowed in Royal Oak?

There is no rule that names barbed wire. Royal Oak's fence chapter contains no list of banned materials at all; no clause mentions barbed wire or electric fences.

That does not make barbed wire approved. Fence materials must be compatible in the judgment of the Chief Building Inspector (City Code § 323-3G), and every fence needs a permit from the Building Department (City Code § 323-6). Approval is a discretionary call made on your application, not a yes-or-no answer printed in the code.

In practice, a residential barbed wire fence would face that compatibility review. Ask the Building Department for its position before you buy anything, and put the question in writing if the answer matters to your project. If the Inspector denies the material, appeals go to the Zoning Board of Appeals (City Code § 323-5C).