What kind of tenancy agreements are acceptable in order to get landlord insurance?

Assured Short Hold

This is the most common type of tenancy agreement, also known as "AST's".  The rented property must be private, the tenancy must have started after 15th Jan 1989, the property must be the tenant's main accomodation and the Landlord cannot live there.  You cannot have an AST if the rent is more than £100k per year or if the property is entirely rent free.  You cannot have an AST if the rent is less than £250 per year or £1000 per year in London.  It cannot be a business tenancy or a tenancy linked to licenced premises.  It cannot be a holiday let.  The Landlord cannot be the local authority.

Assured

Likely to have started pre Feb 1997.  Similar rights to AST, but the tenant can stay in the property for an unspecified period of time.  There is no automatic right for the Landlord to repossess the property at the end of the tenancy.  They must show the court they have good reason for wanting possession, using one of the grounds of possession in the legislation.

Company Let

When a company takes on a residential tenancy agreement as the 'tenant' rather than the individual.  Common amongst the expatriate community as local employers are usually responsible for their own housing.  These types of let are decreasing as companies insist employees gain agreements in their own name.

Periodic

This automatically follows an AST if the parties do nothing i.e. they don't sign another AST, the tenancy is based on the original agreement and the same clauses etc.  The period of the tenancy depends on the rent payment schedule, so if it's paid monthly it becomes a Monthly Periodic Tenancy (also known as a rolling contract).

Non Housing Act

A contract drawn up between the parties and the content does apply.  The tenant has less protection, they generally expire at the end of a fixed term and there is no notice of termination.  If the fixed term ends and tenancy continues then formal notice of eviction would be required, normally one month.  This option is not very common.