Particle number fluctuations, no matter how small, are present in experimental setups. One should
rigorously take these fluctuations into account, especially for entanglement detection. In this context,
we generalize the spin-squeezing inequalities introduced by G. Tóth et al. [Phys. Rev. Lett. 99,
250405 (2007).]. These new inequalities are fulfilled by all separable states even when the number
of particles is not constant and may present quantum fluctuations. These inequalities are useful for
detecting entanglement in many-body systems when the superselection rule does not apply or when
only a subspace of the total system Hilbert space is considered. We also define general dichotomic
observables for which we obtain a coordinate-independent form of the generalized spin-squeezing
inequalities. We give an example where our generalized coordinate-independent spin-squeezing in-
equalities present a clear advantage over the original ones.