Major open problems on polynomial kernel (non) existence
Currently, I would say the 3 major open cases are:


Directed feedback vertex set (make a given digraph acyclic by deleting at most k vertices) parameterized by the size of the solution
Planar Vertex Deletion (make a graph planar by deleting at most k vertices)
Edge Multiway cut (given an undirected graph and a list of terminals, delete at most k edges to ensure all the terminals end up in a different connected component)


For all of these, the relevant parameter is the size of the solution. You can have a look at the open problem list from the 2013 Workshop on Kernelization ( http://worker2013.mimuw.edu.pl/slides/worker-opl.pdf ) for others. Pointers to other (but older) open problem lists in parameterized complexity can be found here: http://fpt.wikidot.com/open-problems .