Modified dispersion relations are a powerful tool for effectively capturing the impact of quantum gravity on particles and fields. They describe the propagation of particles and fields in quantum spacetime by modelling their interaction with the fundamental constituents of quantum spacetime (gravitons) in terms of a four-momentum-dependent background geometry.

The goal of this talk is to demonstrate the influence of modified dispersion relations on observables that can be scrutinised through observations. Around black holes, I will discuss their impact on the black hole shadow, gravitational lensing, and the Shapiro delay. On cosmological scales, I will address the emergence of time delays in the arrival times of simultaneously emitted gamma-ray photons of different energies. I will also briefly discuss modifications in the lifetimes of unstable elementary particles.

To this end, I will present how modified dispersion relations emerge in quantum-gravity phenomenology and in the description of wave propagation through media. It will become clear that dispersion relations can, in general, be interpreted as Hamilton functions determining the propagation of massive and massless point particles. Modified dispersion relations are encoded in Hamilton functions that are not quadratic in the four-momentum of the particle (as is the case in general and special relativity). These modified Hamilton functions will serve as the starting point for deriving the aforementioned observables.