FAA-approved balloon construction is governed by Federal Aviation Regulation (FAR) Part 31. Nothing in Part 31 says balloons must be spheroidal however common sense, practical experience, experiments, engineers, and computers dictate the "natural shape" is best for many reasons.
The word "shape", although not an official technical balloon term, has come to represent balloons that are not spheroidally shaped. To those who care, a sometimes- very-subtle distinction is made between "real" shapes (such as polar bears, trousers, and motorcycles) and spheroids with inflated appendages, such as mouse, bear or football player heads.
Almost 20 years ago, in England, we watched a pick-up pilot and a pick up crew try to inflate the "Levi" balloon (shaped like a pair of pants), only to find they could get only one leg inflated. About that same time, a pilot flying a spark-plug-shaped balloon, after a difficult landing, ran screaming across a field, "I’m never going to fly that (expletive) balloon again."; and we heard that no pilot made more than one flight in that balloon. The technology of designing and building shaped balloons has become considerably more sophisticated recently, but there are some shapes that simply do not lend themselves to safe flight.
Cameron Balloons (US and England) and Aerostar have built most of the "shape" balloons flying in the US, and have received as "standard" certification from the Federal Aviation Administration (FAA) for many of their shaped balloons.
Since the majority of "shape" balloons have been manufactured by Cameron Balloons Ltd. in Bristol, and many fly in the US with British registration, there has been discussion from time to time regarding British vs. US regulations. British Civil Airworthiness Requirements (BCAR) Part 31, is based on US FAR Part 31, and is so acknowledged in the BCAR. There are some differences, however, both regulations require testing of each shaped balloon, both regulations have exactly the same wording on Controllability-"The applicant must show that the balloon is safely controllable and maneuverable during take-off, ascent, descent, and landing without requiring exceptional piloting skill"-which pretty much moots any meaningful discussion about which regulation produces better, safer balloons. As with natural shape balloons, it’s the quality of the company building the balloon that’s important.
Spheroids with inflated appendages may require some special attention during inflation and deflation, but generally fly like standard spheroidal balloons. Real "shape" balloons often have non-standard deflation ports, inflated sections which require special attention during inflation and deflation, internal catenaries and cords which must be correctly deployed, and may require a large crew familiar with the balloon.
Any repair facility certificated to work on a particular manufacturer’s products may work on "shape" balloons made by that manufacturer. Shapes can present problems for repair personnel. Shapes are usually big, heavy, and complicated, and often have no special repair instructions. They sometimes use materials that are not common to regular balloons and, consequently, are not likely to be in repair station stock.
Since many real "shape" balloons are owned by companies and used strictly for advertising, maintenance is often performed while the balloon is on the road (usually with a tight schedule, especially during summer months), repairs are done in a rush, and the balloons don’t get the tender loving care bestowed on most privately-owned aerostats.
Some famous shapes are so unwieldy they shouldn’t carry passengers, be flown in crowds of people, or over congested areas, but that’s their Job.