This paper summarizes the body of work that we have done over the years on the oscillation processes in sunspots, including their umbra, penumbra, and close vicinity. The study analyzes a number of aspects that impede adequate determining of some characteristics of propagating oscillations and lead to misinterpretation. Using running penumbral waves as an example, we show that their horizontal propagation with decreasing frequency is delusive. The effect is due to different oscillations propagating along magnetic field lines with gradually increasing inclination. This also applies to the three-minute oscillations in the sunspot umbral chromosphere. The change in the inclination of the strips in the half-tone space-time diagrams, which are employed to determine the oscillation propagation velocities along coronal loops, is caused by the projection effect as opposed to real changes in the velocity. We propose to use flare modulation of the natural oscillations of the medium to eliminate the uncertainties that arise while measuring the phase differences between signals of the same parameters, which is employed for estimating wave propagation velocities in the solar atmosphere.