Lately, novel porous materials called metal-organic frameworks (MOFs) emerged and immediately attracted attention of the scientific community. These crystalline compounds are unique due to the high degree of porosity which can be utilized for gas adsorption related applications. Additionally, many MOFs containing paramagnetic transition-metal ions exhibit peculiar magnetic properties. The organic part in several of such compounds consists of polar molecules which below a certain phase transition temperature order into a ferroelectric-type phase, making these materials single-phase hybrid multiferroics. We use EPR, ENDOR and dielectric spectroscopic techniques to investigate and characterize the ferroelectric-like phase transitions in manganese and copper doped dense [A][Zn(HCOO)3] MOFs based on the perovskite architecture. Here A+ is NH2NH2CH3+ or (CH3)2NH2+) molecular cations. The temperature dependent continuous-wave EPR spectra reveal that the local paramagnetic ion-probes are indeed sensitive to the local structural changes occurring at the phase transitions. Spectral simulations are used to obtain the g, hyperfine A and fine structure D tensors at different temperatures. This allows us to probe the temperature dependence of the local order parameter and to characterize the observed phase transitions. Pulse EPR and ENDOR measurements are performed to study structure of the framework, lattice dynamics and motion of the molecular cations in the low temperature phases. The magnetic resonance methods are complemented by the dielectric spectroscopy of MOF single crystal samples providing information about the nature of the phase transitions and dynamics of the molecular cations.