  We present detailed derivations, various improvements and application to
concrete experimental data of spin squeezing inequalities formulated recently
by some of us [Phys. Rev. Lett. {\bf 95}, 120502 (2005)]. These inequalities
generalize the concept of the spin squeezing parameter, and provide necessary
and sufficient conditions for genuine 2-, or 3- qubit entanglement for
symmetric states, and sufficient entanglement condition for general $N$-qubit
states. We apply our method to theoretical study of Dicke states, and, in
particular, to $W$-states of $N$ qubits. Then, we analyze the recently
experimentally generated 7- and 8-ion $W$-states [Nature {\bf 438}, 643
(2005)]. We also present some novel details concerning this experiment.
Finally, we improve criteria for detection of genuine tripartite entanglement
based on entanglement witnesses.
