We present a complete analysis of multipartite entanglement of three-mode
Gaussian states of continuous variable systems. We derive standard forms which
characterize the covariance matrix of pure and mixed three-mode Gaussian states
up to local unitary operations, showing that the local entropies of pure
Gaussian states are bound to fulfill a relationship which is stricter than the
general Araki-Lieb inequality. Quantum correlations will be quantified by a
proper convex roof extension of the squared logarithmic negativity (the
contangle), satisfying a monogamy relation for multimode Gaussian states, whose
proof will be reviewed and elucidated. The residual contangle, emerging from
the monogamy inequality, is an entanglement monotone under Gaussian local
operations and classical communication and defines a measure of genuine
tripartite entanglement. We analytically determine the residual contangle for
arbitrary pure three-mode Gaussian states and study the distribution of quantum
correlations for such states. This will lead us to show that pure, symmetric
states allow for a promiscuous entanglement sharing, having both maximum
tripartite residual entanglement and maximum couplewise entanglement between
any pair of modes. We thus name these states GHZ/$W$ states of continuous
variable systems because they are simultaneous continuous-variable counterparts
of both the GHZ and the $W$ states of three qubits. We finally consider the
action of decoherence on tripartite entangled Gaussian states, studying the
decay of the residual contangle. The GHZ/$W$ states are shown to be maximally
robust under both losses and thermal noise.