  We study the interplay of confining potential, electron-electron interaction,
and Zeeman splitting at the edges of fractional quantum Hall liquids, using
numerical diagonalization of finite-size systems. The filling factors studied
include 1/3, 5/2, 2/5, and 2/3. In the absence of Zeeman splitting and an edge,
the first two have spin fully polarized ground states, while the latter two
have singlet ground states. We find that with few exceptions, edge
instabilities of these systems are triggered by softening of edge spin waves
for Abelian fractional quantum Hall liquids (1/3, 2/5 and 2/3 liquids), and are
triggered by softening of edge magnetoplasmon excitations for non-Abelian 5/2
liquid at the smoother confinement side. Phase diagrams are obtained in the
accessible parameter spaces.
