Number of possible cycles in collatz conjecture I had been following all the blogs, but I would like to understand, whether an attempt has been made to understand how many cycles are possible apart from the 1-4-2-1 cycle in collatz problem
 A: A cycle other than the $1-4-2-1$ cycle has not been found. If such a cycle was found, then the conjecture would be disproved. 
If it was proven that no such cycles exist, then the conjecture would still not be solved, since there could be initial values of $n$ for which the recursion diverges.

This is an open problem. Some results regarding cycles of specific types have been proven. 
A: Proofs which deal with the conjecture in terms of bounding the possible number of cycles are not known to me (for instance I've not seen such a concept referred to in Lagarias's survey), but you might take a deeper look (than me) at Kurt Mahler's work of z-numbers; he proves, that only finitely many z-numbers exist and this is known to be related to the cycle-problem in the Collatz-map. There is a bit about that z-numbers on mathworld, and the Mahler-article is available in some online digitized library.        
Besides that it is with elementary means possible to prove, that cycles of some specific lengthes cannot exist, that means one could look at the length of a (virtual) list of excluded lengthes, but I doubt that someone would have tried to make an argument that such a list of excluded cycle-lengthes has some bound on its own length or is either finite or infinite. (for some finite lengthes to be excluded you can see into my amateurish article in the section of "general cycles").           
It is a topic in already early articles (I think R. Terras was the first one) to prove that if one cycle exists at all, it must have a length of at least so-and-so-many steps (I think it is about 170000 or something - but again: this is about the length of possible cycles, not the upper/lower bound of the possible number of cycles)
