We study the entanglement properties of quantum hypergraph states of $n$ qubits, focusing on multipartite entanglement. We compute multipartite entanglement for hypergraph states with a single hyperedge of maximum cardinality, for hypergraph states endowed with all possible hyperedges of cardinality equal to $n-1$ and for those hypergraph states with all possible hyperedges of cardinality greater than or equal to $n-1$. We then find a lower bound to the multipartite entanglement of a generic quantum hypergraph state. We finally apply the multipartite entanglement results to the construction of entanglement witness operators, able to detect genuine multipartite entanglement in the neighbourhood of a given hypergraph state. We first build entanglement witnesses of the projective type, then propose a class of witnesses based on the stabilizer formalism, hence called stabilizer witnesses, able to reduce the experimental effort from an exponential to a linear growth in the number of local measurement settings with the number of qubits.