  We computationally study the spontaneous phase separation of ternary fluid
mixtures using the lattice Boltzmann method, both when all the surface tensions
are equal and when they have different values. Previous theoretical works
typically rely on analysing the sign of the eigenvalues resulting from a simple
linear stability analysis, but we find this does not explain the fluid
morphologies observed. Here, by combining systematic computer simulations over
the full range of the composition space and theoretical analysis on the
eigenvalues and eigenvectors of the unstable modes, we identify four
fundamental phase separation pathways. In particular, we highlight a dominant
but so-far overlooked mechanism involving enrichment and instability of the
minor component at the fluid-fluid interface
