The first one is whether you work for several companies or principally one. This is a very difficult area without any absolutes. Consequently most agencies operate on the side of caution and require that you work through a limited company. Self-employment through a partnership is very similar to Sole Trader, except that two or more persons are involved. These days it may be difficult to prove to the Inland Revenue that the domestic partner is also a business partner.
Some contractors band together to form a limited company where one is a director.
Variations of this umbrella arrangement exist and some agencies offer some form or other of this approach. As a director you, of course, share responsibilities for the acts of your employees so you may find that something done by another contractor who is also a director of the company could have a serious financial impact for you. Another alternative is as an Agency employee. Some companies after contract staff will only use agency staff. To get the contract work will mean that you become employed by the Agency.