(2-part post):
Ryanair will only accept a passport or driving licence:
http://www.ryanair.com/site/EN/faqs.php?sect=p id&quest=photoid
easyJet, however, will accept any form of photographic ID, such as a bus pass or student ID card. That information isn't shown (in any degree of detail) on their website. However, I know it to be correct because a friend of mine (who has no passport and only the old-fashioned paper driving licence) asked me to check for him. After pressing loads of buttons on easyJet's automated telephone system, I reached a recorded message that made it clear that they'll accept any form of photo ID from a public body (such as a council or educational institution) or from any other recognised body. (I.e. basically anything which isn't blatantly 'home-made'). My friend travelled from Stansted to Dublin, using his senior citizen's bus pass, with no problems.
BmiBaby state this:
"bmibaby accepts all forms of photographic identification, however passengers must ensure that all personal identification is of a good standard. If identification appears to be fraudulent, passengers may be refused for travel.":
http://www.bmibaby.com/bmibaby/en/index.aspx?p =724