Context: The dynamic atmosphere of the Sun exhibits a wealth of magnetohydrodynamic (MHD) waves. In the presence of strong magnetic fields, most spectacular and powerful waves evolve in the sunspot atmosphere. Allover the sunspot area, continuously propagating waves generate strong oscillations in spectral intensity and velocity. The most prominent and fascinating phenomena are the 'umbral flashes' and 'running penumbral waves' as seen in the sunspot chromosphere. Their nature and relation have been under intense discussion in the last decades.Aims: Waves are suggested to propagate upward along the magnetic field lines of sunspots. An observational study is performed to prove or disprove the field-guided nature and coupling of the prevalent umbral and penumbral waves. Comprehensive spectroscopic observations at high resolution shall provide new insights into the wave characteristics and distribution across the sunspot atmosphere. Methods: Two prime sunspot observations were carried out with the Dunn Solar Telescope at the National Solar Observatory in New Mexico and with the Vacuum Tower Telescope at the Teide Observatory on Tenerife. The two-dimensional spectroscopic observations were performed with the interferometric spectrometers IBIS and TESOS. Multiple spectral lines are scanned co-temporally to sample the dynamics at the photospheric and chromospheric layers. The time series (1 - 2.5 h) taken at high spatial and temporal resolution are analyzed according to their evolution in spectral intensities and Doppler velocities. A wavelet analysis was used to obtain the wave power and dominating wave periods. A reconstruction of the magnetic field inclination based on sunspot oscillations was developed. Results and conclusions: Sunspot oscillations occur continuously in spectral intensity and velocity. The obtained wave characteristics of umbral flashes and running penumbral waves strongly support the scenario of slow-mode magnetoacoustic wave propagation along the magnetic field lines. Signatures of umbral flashes and running penumbral waves are found already in the middle to upper photosphere. The signal and velocity increases toward the chromosphere. The shock wave behavior of the umbral flashes is confirmed by the evolving saw-tooth pattern in velocity and the strong downward motion of the plasma right after the passage of the shock front. The power spectra and peak periods of sunspot waves vary significantly with atmospheric altitude and position within the sunspot. In the vertical field of the umbra, the mixture of wave periods in the lower photosphere transforms into a domination of the 2.5min range in the upper photosphere and chromosphere. In the differentially inclined penumbra, the dominating wave periods increase with radial distance. The acoustic cut-off frequency which blocks the propagation of long-period waves is considered to increase with the field inclination and the ambient sound speed. The reconstruction of the sunspot's magnetic field inclination based on the peak period distribution yields consistent results with the inferred photospheric and extrapolated coronal magnetic field.