We study the effect of sub-grid physics, galaxy mass, structural parameters and resolution on the fragmentation of gas-rich galaxy discs into massive star-forming clumps. The initial conditions are set up with the aid of the ARGO cosmological hydrodynamical simulation. Blast-wave feedback does not suppress fragmentation, but reduces both the number of clumps and the duration of the unstable phase. Once formed, bound clumps cannot be destroyed by our feedback model. Widespread fragmentation is promoted by high gas fractions and low halo concentrations. Yet giant clumps M > 108 M⊙ lasting several hundred Myr are rare and mainly produced by clump–clump mergers. They occur in massive discs with maximum rotational velocities V max > 250 km s−1 at z ∼ 2, at the high-mass end of the observed galaxy population at those redshifts. The typical gaseous and stellar masses of clumps in all runs are in the range ∼107–108 M⊙ for galaxies with disc mass in the range 1010–8 × 1010 M⊙. Clumps sizes are usually in the range 100–400 pc, in agreement with recent clump observations in lensed high-z galaxies. We argue that many of the giant clumps identified in observations are not due to in situ fragmentation, or are the result of blending of smaller structures owing to insufficient resolution. Using an analytical model describing local collapse inside spiral arms, we can predict the characteristic gaseous masses of clumps at the onset of fragmentation (∼3–5 × 107 M⊙) quite accurately, while the conventional Toomre mass overestimates them. Due to their moderate masses, clumps which migrate to the centre have marginal effect on bulge growth.