There is no requirement for sole traders to register a business name. They can trade under their own names or choose a different business name. However, if as a sole trader you choose something other than your own name for the business, you will have to include your own name and the business address on all business letters, orders and receipts. Invoices must state your VAT registration number if applicable.
Partnerships must state on letters, orders, invoices and receipts the names of all partners and the address of the principal office or alternatively indicate where a list of partners may be inspected. Invoices must state your VAT registration number if applicable.
Limited companies must show the name of the company, the country of registration, the company registration number, the address of the registered office and the address of its place of business, if different. There is no need to list the names of the company directors, but if you do, all must be shown. Invoices must show at least the company name and VAT number if applicable.