1) It is against Federation law to keep a breeding tribble without a valid breeder's licence and a current membership in one of the tribble associations.
2) Tribbles may not be sold, traded, or otherwise change owner without a Non-Breeding Tribble Certificate (NBTC). The NBTC should be complete, including some hair as a DNA sample. Breeding tribbles may only be sold to and by individuals with a valid breeder's licence and a current membership in one of the tribble associations.
3) A non-breeding tribble, accompanied by an NTBC, may be transported within the Federation. However, some planets have their own local laws pertaining tribbles, and not all Captains allow tribbles aboard their vessels, so it is necessary to make inquiries before taking a tribble on a move or vacation.
4) A breeding tribble is considered a potentially harmful life form, and the owner carries certain responsibilities as to the safety and prevention of escape. Licensed individuals are allowed to travel with a breeding tribble, but many ships and planets have their own laws and rules here, as well.
5) Any breeding tribble must be kept in an enclosed environment conformed to a set of standards as to safety and prevention of escape.
6) Tribbles are not susceptible to many diseases, but as any life form, they are possible carriers of such. Some planets require health certificates, tests, and/or quarantine, as do some ship's Captains. Here again, it is a good idea to check before taking your tribble along.
7) It is forbidden for a Klingon to own a live tribble. The Humane Society has ruled it would be extremely stressful for the tribble. As it would also be rather stressful for the Klingon, many feel this regulation is superfluous.
8) And of course under no circumstances should a live tribble be taken into the Klingon empire.