I ran about 30 successful trials over the weekend, about a 2:1 BAF to Lambda ratio, and it looked to me like an even split bet. common, uncommon & rares w/just a few peeks at the very rare (2). Actually, maybe the rare table is a bit more scarce as well--I'm sitting on a lot of unused uncommons atm which I'll likely break into threads (8-10, iirc).
I started out trying to plan out which components to take too, but considering how many trees are out there, I'm basically taking one of each, other than for very rares and commons. Very rares, cuz I've only gotten 2 so it's easy enough to plan which T4 I want, and commons which are ironically "rare" enough that I can see which slot I'm crafting next to pick.
An Offensive Guide to Ice Melee
I got the uncommon table once and the rare table once (the only reason I have a tier 3 in Interface) but the other 5 times I got the common table.
I also got a an uncommon dropped on me once during a Lambda outside of the reward tables. I have no idea what I did to deserve that but it did determine which way I went with my tier 2 in Interface. 
You get a free uncommon component when you get a badge in the trial. When you get the Masta Badge, you get a free rare, IIRC. After you have got the badge, recompleting the badge req's grants a free astral merit
I think it's based on APM exclusively.
Ok, maybe not. The idea of 24 CoH players frantically hitting their keyboard keys like those Starcraft 2 pro players makes me smile.
(Let's try to post something actually relevant now.)
There's another thread on the same topic in the "For Fun!" section, over here. It might be better to gather all the information in one place.
I ran over twenty BAFs and a couple Lambdas yesterday, succeeded all but once, and got exactly one table that wasn't uncommon. It was very rare, which was nice! Rather a lot of people have been joking about how common the uncommon table is in-game.
I've taken the insps too, when I realized I had at least five of every uncommon incarnate salvage. Mostly gotten the blue ones, which is a letdown seeing as at that level of play I am not exactly hurting for recovery.