I like all types of percolation models but am especially partial to long-range percolation. At the moment I am working on a problem involving the one-arm exponent of high-dimensional long-range percolation. In the past I have worked a lot on high-dimensional percolation. My PhD thesis is all about the high-dimensional incipient infinite cluster.
A topic I am currently looking into is hypercube percolation. Hypercube percolation is interesting for many reasons, one of them being that it behaves like an interpolation between percolation on the lattice and the Erdos-Renyi random graph (i.e., percolation on the complete graph).