  We show that the rich structure of multipartite entanglement can be tested
following a device-independent approach. Specifically we present Bell
inequalities for distinguishing between different types of multipartite
entanglement, without placing any assumptions on the measurement devices used
in the protocol, in contrast with usual entanglement witnesses. We first
address the case of three qubits and present Bell inequalities that can be
violated by W states but not by GHZ states, and vice versa. Next, we devise
'sub-correlation Bell inequalities' for any number of parties, which can
provably not be violated by a broad class of multipartite entangled states
(generalizations of GHZ states), but for which violations can be obtained for W
states. Our results give insight into the nonlocality of W states. The
simplicity and robustness of our tests make them appealing for experiments.
