The Louisiana Revised Statutes allow local municipal authorities to require registration of bicycles.
SUBPART B. POWERS OF LOCAL AUTHORITIES
§41. Power of local municipal authorities
A. Except as otherwise provided by law, this Chapter shall not be deemed to prevent local municipal authorities, with respect to highways other than state maintained highways within their corporate limits, from adopting ordinances:
(7) Requiring the registration and licensing of bicycles, including the requirement of a registration fee;
New Orleans requires the registration of all bicycles. You can check to see if your parish requires this at Municode.
Sec. 154-1403. - Registration required.
(Code 1956, § 38-192)
(b) This charge becomes effective as of February 1, 1987.
(c) Exemption for the payment of fees shall be for all bicycles under the cost of $100.00.
(Code 1956, § 38-193)
(a) The superintendent of police or any police officer assigned such duty by him, or the superintendent of fire or any employee of the department of fire when assigned such duty in writing by the mayor, upon receiving proper application therefor, may issue a bicycle registration plate to the owner thereof. Each new owner shall register such bicycle in accordance with the provisions of this article. They shall not register any bicycle when they know or have reasonable grounds to believe that the applicant is not the owner of or entitled to the possession of such bicycle.
(b) The superintendent of police and the superintendent of fire shall keep a record of the number of each registration, the date issued, the name and address of the person to whom issued and the number on the frame of the bicycle for which issued and a record of all bicycle registration fees collected by them.
(c) It shall be the duty of the superintendent of fire to forward a true copy of the record of each registration made by him, or any employee of the department of fire, to the superintendent of police.
(d) The superintendent of police may authorize bicycle dealers to register and attach registration plates to bicycles. All such persons authorized to register and attach registration plates to bicycles shall forward immediately to the superintendent of police true copies of such registration. It shall be a misdemeanor to fail to furnish such copies or to furnish false or fraudulent records to the superintendent of police.
(Code 1956, § 38-194)
The superintendent of police or other authorized person upon issuing a bicycle registration shall also issue a registration plate bearing the registration number assigned to the bicycle, and the name of the city. The superintendent of police or other authorized person shall cause such registration plate to be firmly attached to the frame of the bicycle for which it has been issued in such a position as to be plainly visible. No person shall remove a registration plate from a bicycle during the period of ownership for which it is issued except in the event the bicycle is dismantled and no longer operated upon any street in the city.
(Code 1956, § 38-195)