I am a sole trader, but as i understand it (and I may be corrected here!), you can't be a director unless you have a limited company. Once you have a limited company, you are no longer a sole trader even if there is just you! I think thats right, but check it out with an accountant.
I don't think that was the question, SD.
Yes, you can be a sole trader of your own business and also be a director of numerous other limited companies.