This work studies the case where dark matter emerges from a complex scalar field charged under a $U(1)$ global symmetry, which is spontaneously broken. The analysis considers different explicit symmetry breaking terms, motivated by discrete symmetries, and the results show that in some regions of the parameter space these scenarios may be distinguished by combining different observables, such as direct detection and collider signatures. It is also discussed the case where the stabilising symmetry could be spontaneously broken.