This is the city of Tempe, AZ bicycle code which requires bicycle registration - see sec. 7-11.
http://www.tempe.gov/citycode/07bicycles.htm

A few comments on it:
1. Registration uses a different number than the S/N and this number must be chisled/stamped into the frame. Its costs $8 or so. Several LBS have the forms and dies to do this, but the LBS where I bought my bike highly recommended I don't as the place they are told to do it is on the bottom of the BB and this will be a place for rust to develp and they said it would likely void the frame warrantee. I ran into dead ends when I tried to get it engraved with police chief approval instead.
2. While it is required, note sec 7-4 which says there is no penalty or civil sanction for not following 7-11. But one must do 7-11 to not violate 7-12.
3. No thanks.

Here is the relevant texts with my bolding:
"Sec. 7-11. Registration requirements.
(a) Every owner of a bicycle, before the same shall be operated on any of the streets, alleys, sidewalks or public highways in the city, shall register the bicycle with an authorized agent of the city.
(b) This article shall apply to every bicycle owner who has resided in the city for thirty (30) days or longer regardless of whether they are a part-time or full-time resident. In the event that Arizona State University requires registration of bicycles, any bicycle so registered shall be exempt from the registration requirements of this article.
(c) It shall be unlawful for any person to operate a bicycle requiring registration and not registered pursuant to this article on any street, alley, sidewalk or public highway in the city.
(d) Every authorized agent shall, prior to selling a new or secondhand bicycle to anyone residing within the city, ensure that the bicycle is registered pursuant to this chapter. Dealers of new or secondhand bicycles who have not been designated as "authorized agents" shall, prior to selling same to anyone residing within the city, inform the purchaser of the requirements of this chapter.
(e) This article shall in no way interfere with the secondhand dealer's responsibility pursuant to chapter 16, Tempe City Code.

Sec. 7-12. Bicycle registration cards—Required information.
The bicycle registration cards will contain the following information:
(a) Name and address of owner;
(b) Make, model, serial number, license tag number, color and general description of bicycle;
(c) Date issued;
(d) Change of address and new owner information.

Sec. 7-13. Same—Distribution.
Repealed.

Sec. 7-14. Fees.
The fee for registration of each bicycle shall be set by council (See Appendix A) and be collected and retained by the authorized agent of the city to defray the costs of registration. This fee may be waived by the police or fire chief or their designees for registrations made pursuant to a city-sponsored bicycle registration program.

Sec. 7-15. Duration.
(a) The registration provided for in this article shall be valid for the life of the bicycle. All bicycle registrations shall be appurtenant to the specific bicycle for which issued, and no other, and shall not be transferred to or used on any other bicycle.
(b) It shall be the duty of every person who sells or transfers ownership of any bicycle to report such sale or transfer by returning to the police department the registration card issued to such person, together with the name and address of the person to whom the bicycle was sold or transferred. Such report shall be made within five (5) days of the date of the sale or transfer. It shall be the duty of the purchaser or transferee of such bicycle to apply for a transfer of registration therefor within five (5) days of the sale or transfer. The original registration tag, which has no expiration date, shall remain on the bicycle and the number reassigned to the new registered owner.

Sec. 7-16. Applications for registration.
Application for bicycle registration shall be made by the owner of the bicycle, or upon his written authorization, to the authorized agent.

Sec. 7-17. Registration tags and cards.
(a) The city shall provide to authorized agents the registration tags or dies for use in registering bicycles, together with registration cards.
(b) It shall be the duty of the authorized agents to maintain, repair or replace damaged dies as necessary.

Sec. 7-18. Destruction.
No person shall remove, destroy, mutilate or alter any registration tag, seal or registration card. Nothing in this section shall prohibit the authorized agent from stamping identifying symbols, letters or numbers on the frame or removable parts of licensed bicycle as in their discretion is practicable. The police chief or his designee may authorize an applicant's engraving of the frame in lieu of a stamp. Such engraving must be approved by the chief or his designee.
Sec. 7-19. Removal, etc., of frame numbers.
No person shall willfully or maliciously remove, destroy, mutilate or alter the number of any bicycle frame.

Sec. 7-20. Reissuing tags.
In the event any bicycle registration tag is lost, destroyed or mutilated so as not to be legible, the owner of such registration tag shall immediately make application to an authorized agent, and shall be issued another tag upon approval of such application by the police chief or his designee upon payment of a fee, if any, set by the council (Appendix A Fee Schedule)."