We present a gravitational wave (GW) analysis of an extensive series of three-dimensional magnetohydrodynamical core-collapse simulations. Our 25 models are based on a 15 progenitor stemming from (i) stellar evolution calculations; (ii) a spherically symmetric effective general relativistic potential, either the Lattimer-Swesty (with three possible compressibilities) or the Shen equation of state for hot, dense matter; and (iii) a neutrino parametrisation scheme that is accurate until about 5 ms postbounce. For three representative models, we also included long-term neutrino physics by means of a leakage scheme, which is based on partial implementation of the isotropic diffusion source approximation (IDSA). We systematically investigated the effects of the equation of state, the initial rotation rate, and both the toroidal and the poloidal magnetic fields on the GW signature. We stress the importance of including of postbounce neutrino physics, since it quantitatively alters the GW signature. Slowly rotating models, or those that do not rotate at all, show GW emission caused by prompt and proto-neutron star (PNS) convection. Moreover, the signal stemming from prompt convection allows for the distinction between the two different nuclear equations of state indirectly by different properties of the fluid instabilities. For simulations with moderate or even fast rotation rates, we only find the axisymmetric type I wave signature at core bounce. In line with recent results, we could confirm that the maximum GW amplitude scales roughly linearly with the ratio of rotational to gravitational energy at core bounce below a threshold value of about 10%. We point out that models set up with an initial central angular velocity of 2π rad s-1 or faster show nonaxisymmetric narrow-band GW radiation during the postbounce phase. This emission process is caused by a low T/|W| dynamical instability. Apart from these two points, we show that it is generally very difficult to discern the effects of the individual features of the input physics in a GW signal from a rotating core-collapse supernova that can be attributed unambiguously to a specific model. Weak magnetic fields do not notably influence the dynamical evolution of the core and thus the GW emission. However, for strong initial poloidal magnetic fields (≳1012 G), the combined action of flux-freezing and field winding leads to conditions where the ratio of magnetic field pressure to matter pressure reaches about unity which leads to the onset of a jet-like supernova explosion. The collimated bipolar out-stream of matter is then reflected in the emission of a type IV GW signal. In contradiction to axisymmetric simulations, we find evidence that nonaxisymmetric fluid modes can counteract or even suppress jet formation for models with strong initial toroidal magnetic fields. The results of models with continued neutrino emission show that including of the deleptonisation during the postbounce phase is an indispensable issue for the quantitative prediction of GWs from core-collapse supernovae, because it can alter the GW amplitude up to a factor of 10 compared to a pure hydrodynamical treatment. Our collapse simulations indicate that corresponding events in our Galaxy would be detectable either by LIGO, if the source is rotating, or at least by the advanced LIGO detector, if it is not or only slowly rotating.