How important is lineage in Buddhism? I practice with the Triratna Buddhist community and while I am personally very comfortable with the organisation and its values, it has been criticised for a lack of lineage. 

Generally speaking how important is lineage in Buddhism? Do some schools take it very seriously and others less so? If a school has a shaky or unclear lineage is that seen as a bad thing? If so then why? What does lineage give Buddhism?
The only truly important lineage is the lineage of the Buddha. Unfortunately, everyone has their own idea of what that means. Even the eightfold noble path (described by the historical Buddha himself as the indication of a true lineage in DN 16) is interpreted differently by different schools. As a result, lineage becomes somewhat important, in that it determines which interpretation of Buddhism you subscribe to; they can't all be right, after all.

Further, even with mutually compatible interpretations, it is unreasonable to suggest that one could progress according to two divergent paths (e.g. become a bodhisatta and an arahant) at once, so choosing a school is likely to be beneficial in terms of maintaining focus. On the other hand, it may be that such schools can work in serial cooperation. It is said (MN 141) that Sariputta and Moggallana worked in this way, Sariputta teaching students up to sotapanna and Moggallana leading them on to arahantship.

The danger I can see in not choosing a school is that one may become a "jack-of-all-trades and master of none"; in other words, wandering from path to path and never really climbing the mountain.