They do have pieces of Sovereign, but they won't study it.
They do study it. Nothing implies that Sovereign was anything more than a very powerful Geth ship (it did EXPLODE... and most of the technology in the galaxy is based on Reaper tech anyway). They have no reason to think otherwise other than Shepard's word. And unlike Anderson, the council is not your buddy.
Really, I think their behaviour in ME2 makes sense... for the most part. The disappearance of human colonies isn't their problem. Particularly human colonies in the Terminus Systems. Sending their fleet in there would cause a war. Which, really, is the last thing Shepard should want anyway.
They're pretty much willing to do for you what they did in the first game: you're a spectre, you can go wherever you want and nobody loyal to the *council* can stop you. Problem is, in the Terminus systems, there aren't a whole lot of people loyal to the council. So what on earth is the council SUPPOSED to be doing that it's not? It could give you more equipment and resources I suppose, but you're working with Cerberus, which they don't like.
Honestly, the only council member who's really an ass is the Turian one. He's tactless, he's racist, he's cruel... but the Asari is pretty kind and fair, and diplomatic and the Salarian is pragmatic, and logical.