The parabolic Anderson problem is the Cauchy problem for the heat equation on the integer lattice with random potential. It is well-known that, unlike the standard heat equation, the solution of the parabolic Anderson model exhibits strong localisation. In particular, for a wide class of iid potentials it is localised at just one point. In the talk, we discuss a natural modification of the parabolic Anderson model on Z, where the one-point localisation breaks down for heavy-tailed potentials and remains unchanged for light-tailed potentials, exhibiting a range of phase transitions. This is a joint work with Stephen Muirhead and Richard Pymar.