Building Regulations can apply to sheds, but many small, simple garden sheds fall outside the practical scope of enforcement because they are low-risk structures, though this depends on size, use and construction. A shed used as habitable space, or one containing electrical or heating installations, is more likely to need to meet relevant Parts such as structure, fire safety and electrical safety.
The Building Regulations apply to building work as defined under the Building Control Act 1990, and technically this can extend to garden sheds and other outbuildings, though the level of practical relevance depends heavily on the size, use and construction of the structure. A small, single-storey, unheated timber shed used purely for storage is generally a low-risk structure and is unlikely to raise significant Building Regulations concerns, but this is a matter of degree rather than a blanket exemption, and it is separate from the question of whether the shed needs planning permission.
Where a shed is larger, is used as habitable accommodation, a home office, or a granny flat, or includes electrical wiring, plumbing, or a heating appliance, the relevant Parts of the Building Regulations become more clearly engaged, for example Part A (structure), Part B (fire safety) and the wiring rules that sit alongside Part D (materials and workmanship) enforced through electrical safety standards. Structures close to a boundary can also raise fire safety separation issues regardless of their exempted development status for planning purposes.
Because there is no simple size threshold that automatically disapplies the Building Regulations, and because planning exemption and Building Regulations compliance are assessed separately, anyone building a larger or more complex outbuilding, or one intended for habitable use, should check with a competent professional or their local authority Building Control Section about which requirements apply.