Can I put a permanent sign at my house in Birmingham?

Effectively no. Under the permanent sign rules for residential areas, a single-family home gets no permanent signage beyond a window sign of 1 sq ft and security signs totaling 2 sq ft (Sign Ordinance § 1.08, Table D). Cloth pennants and banners are prohibited outright (Sign Ordinance § 1.08).

For context, the same table allows more for other residential properties. Multiple-family buildings may carry a 10 sq ft sign up to 4 ft high. Institutions such as churches, schools, and government buildings may carry 20 sq ft when within 500 ft of homes, or 40 sq ft otherwise, up to 8 ft high.

A home business does not change the answer. Home occupations must show no exterior indication of the business, including signs (§ 5.02). If a sign is essential, the only path is a variance from the Board of Zoning Appeals (Sign Ordinance § 2.06).